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(A
RUDIMENTARY
TREATISE
N THlE
CAL CULU So
I NT E GRA L
BY HOMERSHAM COX., BRAO
BARRItSTER-AT-LAW;
AUTHOR OF A " MANUAL OF THlE WIFFERENTIAL CALCULUS."1
LONDON:
JOHN WEALE, 59, HIGH HOLBORN.
LONDONT
GEORIGE WOODFALL AND SON,
ANGEL COURTo SKINNER STREET.
It I
PREFACE.
THE Differential and the Integral Calculus have been established upon entirely different axioms and definitions by the
several founders of those sciences. The primary ideas of
infinitesimals, fluxions, and exhaustions, though their results
coincide, for the simple reason that all pure truth is consistent with itself, are widely diverse in their abstract nature.
In writing, therefore, on the principles of either Calculus, a
difficulty presents itself in the necessity of electing between
systems, each of which has the sanction of high authority
and peculiar intrinsic merits.
This consideration is of especial importance in a "Rudimentary Treatise," which cannot, of course, fulfil the profession of its title without singleness and simplicity of its
fundamental ideas, and an exactness of thought and language,often very difficult of attainment. The choice of methods
in the present work has been determined partly by historical
considerations. The discoverers of new truths usually search
after them by the simplest and most familiar considerations;
and it seems natural to presume that, as far at least as
abstract principles are concerned, the way of discovery is the
easiest way of instruction.
The original idea upon which Newton based the system of
fluxions, regarded a differential coefficient as the rate of
increase of a function. The idea upon which Leibnitz and
the Bernouillis established the Integral Calculus, regarded
an integral as the limit of the summation of an indefinite
number of indefinitely diminishing quantities. The facility
PREFACE.
with which the idea of "Irate" may be conceived and applied
to the science of which Newton was the great founder, and the
similar advantages of the idea of summation in the Integral
Calculus, determined the selection of the first idea as the
basis of the "Manual of the Differential Calculus" by the
present writer, and the second as the basis of the present
treatise.
The value and importance of what is termed by Professor
De Morgan the " summatory" definition of integration, has
been insisted upon by him and others of the most eminent
modern mathematicians; but the present is probably an almost
solitary attempt to establish the Integral Calculus on thi;t
definition exclusively. Throughout the entire range of the
practical applications of the Integral Calculus-to Geometry,
Mechanics, &c.-the idea of summation is solely and universally
applied. The rival definition of the Integral Calculus-as
the inverse of the Differential Calculus-has a merely relative signification, and is, therefore, essential only in analytical investigations of the relations of the two sciences.
But whatever system be adopted for establishing either
calculus must of necessity involve the idea of limits and
limiting values. An unreasonable reluctance has been sometimes exhibited in adopting this idea in elementary treatises,
whereas that it is one by no means difficult to be conceived is
shewn by its adoption in the first ages of mathematics. By
far greater difficulties have arisen from the shifts to whichl
resort has been had to evade it in theorems of which
demonstrations without it are necessarily illogical.
The idea of limits occurs, or ought to occur, much earlier
in the study of exact science than is generally allowed.
This idea is essentially involved in Arithmetic, Euclid, and
Algebra. The laws of operation with recurring decimals
and surds cannot be accurately established without limitsfor in what sense is the fraction - equal to -333...., or
V/2 equal to another interminable decimal, except as the
limits of the two infinite convergent series represented by
the decimals? Euclid's definition of equality of ratios
PREFACE.
(Book V., Def. V.), is made to include incommensurable
ratios by considerations dependent on the method of limits,
which also occurs repeatedly in Book XII. In Algebra,
as the present writer has endeavoured to shew elsewhere
(Cambridge Mathematical Journal, Feb., 1852), an exact
demonstration of the Binomial Theorem must involve the
method of limits. The same remark applies to the operation
of equating indeterminate coefficients and the theorem ao = 1.
Neglect of these considerations involves the writers of some
treatises in obscurities, errors, and inconsistencies, which
bring to remembrance the supposed common origin of the
words "gibberish" and " algebra."*
Throughout the present work, the language of infinites
and infinitely small quantities has been carefully avoided,
partly because they cannot, except by an inaccuracy of language, be spoken of as really existing magnitudes which may
be subjected to analytical operations, partly because the
language of the method of limits is equally concise, and is,
moreover, exact.
That infinity has a real existence must be admitted; for let
us conceive any distance, however great, such that the remotest
known star is comparatively near; we cannot say that space
terminates at that distance. What is beyond the boundary?
A void, perhaps, but still space; so that unless we can
conceive the existence of a boundary which includes all space
within it, and to which no space is external, we are forced
to admit the existence of infinite space. But this admission
is altogether different from that which subjects infinity to
mathematical operations. How is the infinity thus operated
upon to be defined? As a magnitude than which none other
is greater? But by hypothesis it is the subject of analytical
* Algebra.-" Some, however, derive it from various other Arabic words,
as from Geber, a celebrated philosopher, chemist, and mathematician, to
whom they ascribe the invention of this science."-IIutton's Mathematical
Dictionary. Gibberish.-" It is probably derived from the chemical cant,
and originally implied the jargon of Geber and his tribe."-Johnson's
Dictionary.
PREFACE.
operations, and therefore of addition. Add, therefore, some
quantity; the result is greater than this infinity, or the
definition is contradicted. The truth is, that absolute infinity, such as the infinity of space, cannot be intelligibly
conceived on the supposition that anything can be added
to it.
Similar considerations apply to infinitely small quantities.
There is no difficulty in seeing, that of any kind of magnitude the parts may be diminished infinitely, for, however
small a part be taken, it may be divided, and thus smaller
parts are taken. If, then, an infinitesimal quantity, the
subject of analytical operation, be defined to be a real quantity less than any other, the definition may be readily shewn
to be inconsistent with itself.
When, therefore, infinitesimals and infinity are introduced
into mathematical operations, they ought to be regarded not
as having an absolute existence, but merely as the means of
expressing the limits to which results approach, as quantities
in them are continually increased or diminished.
M1. Cournot, in his admirable treatise " Des Fonctions et
du Calcul Infinitesimal" (Paris, 1841), asserts, indeed, that
the infinitesimal method does not merely constitute an ingenious artifice; that it is the expression of the natural
mode of generation of physical magnitudes which increase
by elements smaller than any finite magnitude. But he
does not appear to have anywhere defined what he understands by elements smaller than any finite magnitudes; and
without such a definition it is impossible to investigate his
proposition accurately. If the words of it be interpreted
literally it appears to lead to this dilemma: if the elements
be not magnitudes, the addition of them produces no increase-if they be magnitudes, they cannot be less than any
finite magnitude; for, being magnitudes, they may be divided
into less magnitudes.
With respect to the method of limits, M. Cournot is of
opinion that questions must occur in which it is necessary
to renounce this method, and to substitute for it in language
PREFACE.
vii
and in calculations the employment of infinitely small quantities of different orders. He has not, however, specified any
instance in which the substitution in question is required.
The following demonstrations do not refer directly or
indirectly to different orders of small quantities, nor, indeed,
to small quantities at all; for the use of the term "small,"
in an absolute sense, in mathematics, is objectionable on
account of its inexactness. The limit where greatness ceases
and smallness begins cannot be distinguished. Hence, though
one quantity may be accurately said to be smaller than another,
the former cannot with perfect exactness be said to be necessarily and absolutely small with respect to the latter.
The exclusive adherence to the "summatory" definition
of the Integral Calculus, has rendered it necessary to present
the greater part of the following propositions in a new form,
and scarcely anything here given (except the historical
notices) is compiled from analogous treatises. The first
section contains a popular exposition of the Integral Calculus; and the second a brief account of its history, compiled from one or two cyclopmedias and dictionaries. The
two following sections are probably in a great measure new,
as in them the general principles of integration and the
integration of the fundamental functions are derived from
the definition above referred to. The three short sections
which succeed contain nothing original; but the eighth, on
Rational Fractions, is almost entirely newly written. The
ordinary demonstration of the possibility of resolving a
rational fraction into partial fractions proceeds by the method
of equating coefficients, and is defective in this respectthat it neglects to shew, c priori, that the assumed coefficients have any real existence, and that the equations
determining them do not give impossible or inconsistent
results.
To the kindness of PROFESsoR DE MORGAN, of University
College, London, the Author is indebted for an exact demonstration of the existence of partial fractions corresponding to rational fractions, with denominators resolvable
PREFACE.
into simple factors. Similar obligations have been conferred
by Ma. COHEN, of Magdalene College, Cambridge, by his
analogous demonstration respecting quadratic factors. In
a subsequent part of the section, a method of effecting these
resolutions is proposed, which may, perhaps, save some
labour.
In the ninth section a hint has been taken from Moigno's
edition of Cauchy's " Leyons de Calcul Integral," to generalize in some measure the principles of Rationalization.
In the next chapter the "summatory" definition is extended to Multiple Integrals. The Quadrature of Curves
and the Cubature of Solids are next considered; and a
method, which is probably new, is given, of investigating the
cubature by polar co-ordinates, by considering surfaces to be
generated by the revolution of figures of variable form.
The theories of rectification of curves and complanation
of surfaces have some difficulties which are frequently
evaded by illogical reasoning. In the "Principia," the
method of rectification is based on the fifth Lemma-" the
homologous sides of similar figures are proportional." This
is stated without demonstration, and is intended to be
axiomatic. It assumes, in other words, that if any figure
be drawn to a reduced scale, the linear dimensions of the
corresponding parts are in the ratio of the scale of the
original to that of the copy. Certain Cambridge versions of
Newton's Lemmas, among other mutilations of the original,
have attempted to prove this axiom respecting lengths, by
reference to a proposition respecting areas, of which the
evidence is of a totally different kind.
Some continental writers, amongst whom is M. Cournot,
have thought to avoid all difficulty respecting the fundamental principles of rectification and complanation, by defining curves and surfaces to be respectively polygons and
polyhedrons of indefinitely small sides. But it is, in truth,
a mere postponement of difficulty to invent new definitions
to answer special purposes. The methods of measuring
curves and surfaces, as defined by M. Cournot, are, perhaps,
PREFACE.
to be connected with his views respecting small quantities,
but cannot be considered complete until extended by rigorous
reasoning to surfaces and curves generated by continuous
motion-such as solids of revolution and their sections.
An essay is made in the following pages to establish the
principles of this part of the Integral Calculus on very
simple geometrical axioms, and the formula of complanation
is proved without the usual reference to the inclination of
tangent planes.
A consideration of the integration of functions which become discontinuous or infinite for particular values, appeared
necessary to complete the subject, and an attempt has been
made to elucidate the definition of multiple integrals of
discontinuous functions. In the concluding section, an investigation of some of the properties of the second Eulerian
integral is partly taken from Littrow's " Anleitung zur
hheren Mathematik;" but in the original proofs an important
defect exists, to remedy which, the article on ultimate ratios
of Eulerian integrals has been given. The demonstration
of the fundamental relation between the two kinds of such
integrals is that of Poisson, as given by M. Cournot. Some
remarks are offered on the inexactness of evaluations of the
sine and cosine of an infinite angle.
Several invaluable suggestions of Professor STOKES, the
Lucasian Professor of Mathematics at Cambridge, have been
embodied in the two concluding chapters; and the obligations
thus conferred are acknowledged by the Author with a feeling
of great gratification.
Geometrical representations of analytical theorems have
been frequently introduced for the purpose of illustration, but
not of demonstration; for though the proof of purely analytical theorems of the Integral Calculus is independent of
the extrinsic aid of geometry, they are often remarkably
elucidated by being considered objectively.
CAMBRIDGE, February, 1852.
CONTENTS.
[The Numbers designate Articles referred to.]
SECTION
I. GENERAL NATURE OF THE INTEGRAL CALCULUS.
II EARLY HISTORY.
III. PRINCIPLES OP INTEGRATION.
Definitions, 11-14, 16, 17, 23-25. Limit of functions, 15.
Quadrature of areas, 19. Limits of quadratures, 20, 21. Integral between its quadratures, 22. Integral, a function of its
limits, 26. Sum of integrals with consecutive limits, 27.
Integration inverse of differentiation, 31. Addition and multiplication of integrals, 32, 33. Integration by parts, 34, 35.
Transformation by change of variable, 37, 38. Indefinite
integration, 40. Differentiation of integrals, 41.
IV, FUNDAMENTAL INTEGRALS.
Integration of ax, 42; of -, 43; of logx, 44. Bernouilli's series,
45. Value of s, 46. Integrals of trigonometrical functions,
47-52; of (X2 a2) - &c., 53-59. Methods of integration
classified, 62.
V, ALGEBRAICAL TRANSFORMATION.
(a+-bx+cx +...), 64; (logx)", 68; functions of a+b, 70;
functions of a2+ 2, 71.
VI. INTEGRATION BY PARTS.
VII. FORMULA; OF REDUCTION.
Extension of Bernouilli's series, 81. Integrals of ( - 2)2, 83;
of (A+B x). (+ 2bx + c)-", 85.
VIII. RATIONAL FRACTIONS.
Definitions, 86-88. Every rational fraction may be resolved into
partial fractions, 87-97. Form of result of the resolution, 98.
Resolution by equating coefficients, by substitution, by differentiation, 99-101. Cases of quadratic factors, 102-104.
CONTENTS.
SECTION
IX. RATIONALIZATION.
To rationalize a function partly rational and partly irrational, 108.
Product of a rational and irrational function, 109. Criteria of
integrability, 111.
X. INTEGRATION WITH SEVERAL VARIABLES.
Definition of multiple integrals, 115, 116. Multiple integrals
found by successive integrations, 117. Order of integration
indifferent, 118. Illustration, 119.
XI. QUADRATURE OF CURVES.
Circle, 122. Ellipse, 123. Oblique co-ordinates, 124. I-yperbola, 125. Area between asymptotes, 126, 127. Polar
co-ordinates, 128, 132. Area in terms of length of curve, 134.
Negative co-ordinates, 135, 136.
XII. CUBATURE OF SOLIDS.
General formula, 137. Limits of integration, 138, 139. Solids
of revolution, 141-144. In terms of area, 146. Cubature by
polar co-ordinates, 147-149.
XIII. RECTIFICATION OF CURVES AND COMPLANATION OF SURFACES.
Lemmas, 150, 151. Curve, the limit of a polygon; limit c--,o
arc
152. Rectification, 153. Curved surface the limit of a polyhedron, 154. Complanation, 156.
XIV. DISCONTINUOUS FUNCTIONS.
Errors arising from integration of discontinuous functions by
ordinary rules, 158. Definition of integrals of discontinuous
functions, 159. " Principal" integrals, 160. Conditions of
determinateness, 161. Geometrical illustrations, 162. Multiple
integrals, 164. Definition, 165.
XV. DEFINITE INTEGRALS.
Eulerian integrals, 168-176. dx-ax cosrx, 171.
Sin oo, 172, 173. Differentiation of definite integrals, 174.
cdx, 2at2 cos2cx, 175.
APPENDIX.
Taylor's Theorem.
INTEGRAL CALCULUS.
SECTION I.
GENERAL ACCOUNT OF TIE OBJECTS OF THE INTEGRAL
CALCULUS.
1. AMONGST the most important uses of the Integral Calculus
are its applications to the measurement of the lengths of
curves, the areas of curvilinear figures, the contents of solids
contained by curved surfaces, and the effects of forces. This
Calculus is required in the most important investigations in
every branch of the exact sciences.
2. The names of the Integral and Differential Calculus
sufficiently indicate the distinction between them. The Integral Calculus determines the whole sum or integral magnitude of a quantity of which the differential parts are given.
The Differential Calculus, on the contrary, investigates the
relations of the differential parts of a quantity of which the
integral magnitude is given.
3. The process of Integration is therefore the inverse of
Differentiation; in the same way as Subtraction is the inverse of Addition, Division the inverse of Multiplication,
Evolution the inverse of Involution. But in the same sense
that Integration is the inverse of Differentiation, the latter
operation is the inverse of the former. As, therefore, the
Differential Calculus is defined and investigated irrespectively
of the Integral, so may also the Integral independently of
the Differential. It is an unnecessarily restricted view
which regards the Integral Calculus as a dependent science.
Throughout the following pages its rules will be independently demonstrated; though the close relation between the,,wo Calculi requires careful consideration, for the sake of its
aid in comprehending both subjects, its suggestiveness in.nvestigation, and its test of results by inverse operation.
INTEGRAL CALCULUS.
4. It was said above, that the Integral Calculus determines
the integral magnitude of a quantity from its differential parts.
Now of course this indirect method of measurement would
not be usually resorted to, if a more direct were practicable.
But there are innumerable cases in which direct measurement
is impracticable. The measurement of the lengths of lines
affords a simple illustration. If the lines be straight, the
nethod of measuring them is obvious and direct. It consists
in successive applications of a straight " rule " or standard
of a unit of length (a yard, metre, ell, &c.), along the straight
line to be measured, and ascertaining how many times it contains the unit and known parts of it. But if the line to
be measured be a curve, no such application of a straight
" rule" can be performed; it will coincide with the curve for
no portion of it, however small.
5. A rough way of effecting the required measurement
is, however, readily suggested. A number of points may
be arbitrarily taken in the curve, and be joined, or be supposed to be joined, by dotted lines. Then, if these chords
be measured, their total length is an approximate measure
of the length of the curve.
6. It was long ago perceived, that by diminishing the
lengths of the chords, and increasing their number, the approximation became closer and closer. An improvement in
the method was effected by drawing from the extremities and
intermediate points of the curve, tangents meeting each other
at points in the convex side of the curve, as in the following
diagram. If the curve be stich that the tangent, at any
OBJECTS OF THE INTEGRAL CALCULUS.
point of it, cannot meet it at any other point, the total
lengths of these tangents is less than the length of the
curve. In this way the length of the curve, though it could
not be exactly determined, might at any rate be ascertained
to be less than one, and greater than another, of two quantities; which might be made to differ by a quantity less and
less, as the number of chords and tangents was increased.
So that the error of the approximation would be determined
within closer and closer extremes, as the geometer expended
more and more labour on the mensuration. It is clear,
however, that the length of the curve has some exact value,
which is the LIMIT of the operations above explained; and
the discovery of that exact limit is the solution of a problem
of the Integral Calculus.
7. Again, the area of any plane curvilinear figure is certainly
greater than that of any polygon of straight sides inscribed
in it, and less than that of any such polygon circumscribed.
By increasing the numbers of sides of the circumscribed and
inscribed polygons, their areas are made to differ less and
less. The area of the curvilinear figure lying between them
may thus be determined within any degree of approximation.
For instance, let the area ACB be included by a curve AB,
and two straight lines, AC, CB, at right angles to each other.
It requires little science to perceive that one of the readiest
vays of roughly measuring this area, is to divide it into portions
by lines parallel to AC, but not necessarily equidistant, and
to compute the area of each such portion as if it were a rectangle. Yet this method would give the area of the figure
boufided not by the curve, but by the zigzag dotted line
B S
INTEGRAL CALCULUS.
within or without the figure. The difference between the two
rectilinear figures bounded by the two zigzag lines may be
reduced by increasing the number and diminishing the areas
of the rectangles. Thus the curvilinear area may be deterI IA
mined within a margin of error which may be diminished at
pleasure. This process for determining areas is called the
Method of QUADRATURES.
8. It may happen that this method of approximation suggests the limit to which it tends. The Integral Calculus
differs from the preceding method only in that it substitute
absolute exactness for mere approximation. The curvijt, ai
figure must have some exact area which is the limit of tt(
results of the above operations. If, therefore, that limit ma,
be inferred from them, they lead to the solution of a problew
of the Integral Calculus.
9. Again, one of the most frequent problems of Dynamics
is to ascertain the distance passed over in a given time by
a point moving with continually-varying velocity. If the
point were moving with uniform velocity, the distance described by it in any time could be immediately ascertained.
The approximation to the distance described by a varying
velocity is analogous to the approximations above described,
and consists in supposing the velocity to change not continuously but after intervals, and remain uniform during each
interval. The shorter the intervals, the more nearly does the
distance computed on this supposition approximate to the
real distance described. Let the distances be computed on
the hypotheses, first, that the point retains throughout
OBJECTS OF THE INTEGRAL CALCULUS.
each of the intervals into which its motion is hypothetically
divided, the velocity it actually has at the commencement of
that interval; secondly, that the point has throughout each
interval the velocity it actually has at the termination of that
interval. The first hypothesis evidently gives the distance
traversed too small; the second hypothesis too large, if the
velocity be a continuously-increasing one. By diminishing
the hypothetical intervals, the error of approximation is reduced; and if the limit to which these operations lead can be
found, the result is the solution of a problem of the Integral
Calculus.
10. The principle on which all the above cases depend,
may be stated generally thus:-A quantity is to be measured
which cannot be immediately compared with the unit of measurement. The quantity is therefore divided into several
parts, and it is ascertained of each of these, that it exceeds
one, and falls short of another, of two quantities measurable
by the given unit. The sums of the two series of measurable quantities are the one greater, the other less, than the
whole quantity to be measured.
This process has been continually practised by the most
unskilful as well as the most skilful computers. It is applied
in innumerable cases in the ordinary avocations of life. The
science which from this kind of approximation extracts
rigorous and exact truth, is the INTEGRAL CALCULUS.
The foregoing remarks will probably suffice to show the
student what kind of reasoning may be expected to engage his
attention in this subject. They serve also to render intelligible the following slight sketch of its history.
INTEGRAL CALCULUS.
SECTION II.
EARLY HISTORY OF THE INTEGRAL CALCULUS.
PYTHAGORAS, born about 590 B.c., died about 497 B.c. The
history of his mathematical discoveries rests generally on no
higher authority than that of tradition. The discovery of the
quadrature of the parabola has been ascribed to him, as appears from the following passage in Dr. Hutton's Mathematical Dictionary. In reference to the theorem that the
square on the hypothenuse of a right-angled triangle is equal
to the sum of the squares on the sides, it is remarked, that
" Plutarch even doubts whether such a sacrifice was made
for the said theorem, or even for the area of the Parabola,
which it was said Pythagoras also found out."
EUCLID, who lived about 280 B.c., and about 50 years before
Archimedes, showed, in his 10th Book, that the areas of the
Circle and Polygon inscribed in it are ultimately equal. He
demonstrated that the area of the circle is equal to half the
rectangle contained by the radius and circumference, and thus
found out a problem of Integration. His method is known
as the method of Exhaustions. The first proposition of the
10th Book asserts that, if from the greater of two given
quantities be taken more than its half, from the resukting
remainder more than its half, and so on continually, there
will remain at last a quantity less than either of the given
quantities. By this reasoning, the difference between the
circle and polygon is exhausted, and the circle becomes ultimately equal to the polygon.
ARCHIMEDES, who lived about 250 B.c., investigated the
ratio of the circumference of a circle to its diameter. By
calculating the length of the periphery of a circumscribed
polygon of 192 sides, and an inscribed polygon of 96 sides,
he found that the circumference of the circle is betwE n
3{o and 3-- of the diameter. He left a treatise on tae
Spiral which now bears his name; and determined the relation of the area bounded by that curve to that of the circumscribed circle. To Archimedes is attributed the quadra
EARLY HISTORY.
ture of the parabola, which discovery, however, as appeare
above, has been assigned to Pythagoras also. Let AO be
a portion of a parabola, 0
its vertex, OB a part of its c- A
axis, and AB a straight line
at right angles to it. The
proposition in question,
which is interesting from
its antiquity and intrinsic
importance, asserts that the
area AOB is two-thirds of
the rectangle ACOB. The
student may easily ascertain oafter reading the following
pages, that this result is equivalent to the integration of a
function of the form cx', where c is constant and x variable,
Archimedes showed in his treatise nITE 2;,aa Ka xvhvov,
that the content of a sphere is two-thirds of that of the
cylinder which just contains it; that the surface of a sphere
is four times as great as that of one of its great circles, &0.
CONON, a contemporary of Archimedes, is said to have
invented the spiral which bears the name of the latter, and
to have proposed to him problems respecting it, which were
solved by him.
PAPPUS, who lived towards the end of the fourth century
(about A.D. 380), demonstrated some of the principal properties of the same spiral, by adding together an indefinite
number of parallelograms and cylinders, into which he sup.
posed a triangle and cone ultimately divided. Pappus also
gave in the preface to his 7th Book, the centrobaric method
of determining the content and superficies of a solid of revolution in terms of the dimensions of the generating figure,
and the position of its centre of gravity. The theorems of
the centrobaric method discovered by Pappus, frequently are
called Guldin's properties, from a much later mathematician,
Guldini, by whom they were demonstrated.
GALILEO, born 1564, died 1642, proved that a body
moving in a straight line with a constant acceleration, such
as that produced by gravity, describes in any time from the
commencement of the motion a distance proportional to that
time. He thence showed that the path of a projectile is a
parabola. The determination of the distance described by a
INTEGRAL CALCULUS.
constantly-accelerated point depends necessarily on the principles of the Integral Calculus, as explained in Article 9.
TORRICELLI, born 1608, died 1647, was a disciple of Galileo, and wrote a treatise De Dimensione Parabolce, with an
appendix De Dimensione Cycloidis. Dr. Hutton says, that
Torricelli "first shewed that the cycloidal space is equal to
triple the generating circle (though Pascal contends that
Roberval shewed this); also, that the solid generated by the
rotation of that space about its base, is to the circumscribing
cylinder as 5 to 8; about the tangent parallel to the base, as
7 to 8; about the tangent parallel to the axis, as 3 to 4," &c.
(See DESCARTES.)
CAVALIERI, a disciple of Galileo, and friend of Torricelli,
published in 1635, Geometria Indivisibilibus continuorum
novd quddam ratione promota, 4to., Bononie. This work,
which obtained for the author the credit in Italy of inventing
the Infinitesimal Calculus, proceeds by division of geometrical
figures into indefinitely small parts.
ROBERVAL, in 1646, determined the centres of percussion
and centres of gravity of sectors of cylinders and circles, &c.,
by methods equivalent to Integration. From the letters of
Descartes, it appears that these discoveries were subjects of
controversy between him and Roberval. Roberval's Treatise
on Indivisibles, appeared in 1666, in the Memoirs of the
Academy of Sciences at Paris.
DESCARTES, born 1596, died 1650, determined the centres
of gravity and centres of oscillation of various curvilinear
figures. His method of demonstrating the proposition respecting the cycloid, referred to in the preceding notice of
Torricelli, is an excellent instance of the geometrical investigation of the quadrature of curves. The following is an
extract from a letter from him to Father Mersenne, in 1638.
(Lettres de Descartes, tome iii. page 384, Paris, 1667.)
" You commence by an invention of Monsieur de Roberval,
respecting the space included by the curve described by a
point of the circumference of a circle supposed to roll on a
plane; with respect to which, I acknowledge that I have
never before thought of it, and that the observation of it is
pretty enough. But I do not see that there is reason to
make so much noise at having found a thing which is so
easy, and which any one who knew ever so little of geometry
could not fail to find if he sought for it. For if ADC be
EARLY HTSTORY.
this curve, and AC a straight line equal to the circumference
of the circle STVX, having divided this line AC into 2, 4,
8, &c., equal parts, by the points B, G, H, N, 0, P, Q, &c., it
S D
p Y
/ E F
X tT
V A N G 0 B P H Q
is evident that the perpendicular BD is equal to the diameter
of the circle, and that the whole area of the rectilinear
triangle ADC is double of this circle. Then, taking E for
the point where the same circle would touch the curve AED,
if it were placed on its base at the point G, and taking also
F for the point where it touches this curve, when it is placed
on the point H of its base, it is evident that the two
rectilineal triangles AED and DFC are equal to the square
STVX inscribed in the circle. Similarly, taking the points
I, K, L, M for those where the circle touches the curve when
it touches its base at the points N, O, P, Q, it is evident
that the four triangles AIE, EKD, DLF, and FMC are
together equal to the four isosceles triangles inscribed in the
circle SYT, TZV, VIX, and XQS; and that the eight other
triangles inscribed in the curve on the sides of these four
are equal to the eight inscribed in the circle, and so on to
infinity; whence it appears that the whole area of the two
segments of the curve, which have AD and DC for bases, is
equal to that of the circle; and, consequently, the whole area
contained between the curve ADC and the straight line AC,
is triple that of the circle."
GREGORY (St. Vincent) of Bruges, published in 1647,
Opus Geometricum Quadrature Circuli et Sectionum Coni.
He showed that the space between a hyperbola and its
asymptote is divided into equal portions by straight lines,
which divide the asymptote into parts in geometrical progression, and which are parallel to the other asymptote.
FERMAT, who died 1663, was author of a "Method for
Quadrature of all sorts of Parabolas," and a treatise on
* By a property of the circle mentioned in the notice of Euclid.
B3
INTEGRAL CALCULUS,
Maxima and -Minima, in which problems concerning the
centres of gravity of solids are solved by a method resembling Newton's Fluxions.
HUYGENs, in 1651, published Tlheoremata de Quadraturda
Hyperbolc, Ellipsis et Circuli ex dato Portionum Gravitatis
Centro; and in 1658, at the Hague, his celebrated Horologiumr Oscillatorium sive de motu Penduloru~m, in which he
states that he was the first discoverer that a certain segment
of the cycloid is equal to a regular hexagon inscribed in the
generating circle. He showed that the time of oscillation of
the cycloidal pendulum is independent of the extent of vibration, and from the principles of the pendulum measured
the effect of gravity, by which he showed that a body
descended vertically from rest in vacuo, in the latitude of
Paris, 15 French feet in one second.
'WALLIS, in 1655, published his Arithmetica Infinitorum, a
great improvement on the Indivisibles of Cavalieri. Wallis
treats of quadratures, and gives the first expression for the
quadrature of a circle by an infinite series in this work,
" in which," says Professor De Morgan, " a large number
of problems of the Integral Calculus is solved, and which
contained more hints for future discovery than any other
work of its day."
NEAL, in 1657, made a remarkable step in the Integral
Calculus. He appears to have been the first person who
determined the exact length of any curve. Wallis, in his
Treatise on the Cissoid, states that Neal's rectification of the
semi-cubical parabola was published in July or August, 1657.
VAN HAURENT, in Holland, in 1659, also gave the rectification of the semi-cubical parabola, as appears from Schooten's
Commentary on Descartes' Geometry.
GREGORY (JAMEs) published, in 1667, Vera Circuli et Hyperbol(e Quadratura, to which he added in the year following
Geometric Pars Un iversalis, of which the method resembles
that of Roberval's Indivisibles.
Dr. BARRow, in 1670, published his Method of Tangents.
He died in 1677, and the year following appeared his demonstrations of Archimedes' properties of the Sphere and Cylinder, by the method of Indivisibles.
LEdIBNITZ, in 1684, gave in the Leipsic Transactions an
account of his Differential Calculus. It is agreed that this
was the first time that this grand discovery appeared in print;
though in the celebrated controversy which arose as to his
EARLY HISTORY.
11,P aim to the priority of this invention, a Committee of the
Rloyal Society decided that " Sir I. Newton had even invented his method before 1669." The general opinion of
modern mathematicians appears to concede to Leibnitz the
merit of an independent discovery, and to exempt him from
the charge of plagiarism.
GrEGORY (DAvrI) published, in 1654, Exercitatio Geometrica de Dimensione Figurarumn.
NEWTON published his Principia in 1687, the most memorable year, therefore, in the annals of science. The doctrine
of limits, conceived and applied in the earliest periods of
mathematical research, had been rapidly growing in importance at the time of Newton and Leibnitz. The great step
made by them consisted in connecting the idea of limits with
a specific notation, and in erecting into a regular system a
science \hich before their time had been exhibited only
in isolated theorems. A large part of the results of the
Priacipi are demonstrated by geometrical miethods equivalent to Integration. Newton's Method of Fluxions was first
published in 1704, subjoined to his treatise on Optics.
MEROiATOR (NICHOLAS), in 1688, published his Logarithmoteclhiia, and is stated to have been the first person who
ever investigated the quadrature of curves analytically. This
he did in a Demonstration of Lord B3rounchlier's Quadrature
of thle Hyperbola, by Wallis's method of reducing an algebraie;d fraction to an infinite series by division.
By thef English contemporaries of Newton, the Integral
Calculus, a Differential Coefficient, and an Integral, were
called the Inverse Method of Fluxions, a Fluxion, and a
Fluent respectively. The notation and phraseology of fluxions
is now almost obsolete. The methods of Exhaustions, Prime
and Ultimate Ratios, Infinitesimals, Indivisibles, Residual
Analysis, Analysis of Derivations or Derived Functions, and
of Limits, are different appellations which the same subject
has at different times received.
From the time of Newton and Leibnitz the Integral Calculus rapidly advanced. Its progress was in a great degree
due to John and James Bernouilli, who published a large
number of memoirs on the subject; to Maclaurin, whose
Fluxions appeared in 1742; to Cotes, whose Harmonia Mensurarum appeared in 1722; to D'Alembert, who gave Memoirs
on the Calculus in the Paris and Berlin Memoirs; and to
12 INTEGRAL CALCULUS.
Euler's great work, Institutio Calculi Integralis. Petr. 1768,
3 vols. 4to.
The analytical part of the Integral Calculus consists in
reducing integrals to forms by which their numerical values
may be computed. This computation is usually facilitated
by the common mathematical tables of sines, cosines, logarithms, &c. But many integrals cannot be found by these
tables. In order to compute such integrals, other tables
have been constructed, of which the principal are called
Tables of Elliptic Integrals, from their relation to the length
of elliptic arcs.
FAGNANO, in his Produzione Matenmatiche, 1750, investigated a remarkable theorem respecting these arcs, which
bears his name, and shows how the length of two arcs may
be taken so as to differ by an assigned algebraical quantity.
EULER gave to the world some of the most important discoveries which constitute the basis of this branch of the Integral Calculus. In 1761 he published, in the Petersburgh
Transactions, the complete integration of an equation involving two terms, each an elliptic function not separately
integrable. Euler also invented the class of integrals which
are known as Eulerian Integrals.
LANDEN, in 1775, published his theorem showing that any
arc of a hyperbola may be measured by two arcs of an
ellipse.
LAGRANGE'S Memoirs in the Turin Transactions, in 1784
and 1785, greatly extended the subject of elliptic functions
in a part of it which Euler had not discussed, and rendered
the determination of numerical values of elliptic functions
very complete.
LEGENDRE undertook the task, involving immense labour,
of computing a greatly-extended series of tables. The second
volume of Legendre's great treatise on elliptic functions, to
which a large part of his life had been devoted, appeared in
1827. To him is attributed the merit of giving to the subject that systematic arrangement and connection which constitute it a separate science.
JACOBI, Professor of Mathematics in Koningsburg, published shortly afterwards, in Schumacher's Journal, his researches on elliptic functions. His principal object was the
investigation of certain general relations of these functions,
of which the investigations of Lagrange and Legendre involve
particular cases.
EARLY HISTORY.
13
ABEL, Professor of Mathematics in Christiania, gave investigations of the subject in Crelle's Journal, in 1827. He
arrived independently at many of the important discoveries
of Jacobi, and contributed valuable theorems respecting what
are called ultra-elliptic functions. The works of Abel, who
died at the early age of 27 years, are esteemed among the
most important contributions to modern analysis.
For some account of modern discoveries in Calculus, the
reader may be referred to Moigno's edition of Cauchy's LeVons
de Calcul Diff6rential et de Calcul Integral, 1814.
Among the best known general works on the Integral Calculus are the following:Bossut, Cal. Diff. et Integral. Paris, 1798.
lBouchlrlat, Differential and Integral Calculus, Eng. Translation. Cambridge, 1828.
Carnot, Metaphysique de Calcul Infinitesimal. Paris, 1796.
Cauchy, Lecons de Cal. Diff. et Int. Vol. 2, Calcul Integral. Paris,
1844.
Condorcet, Calcul Integral. Paris, 1765.
Cournot, Des Fonctions et du Calcul Infinitesimal. Paris, 1841.
De Morgan's Diff. and Integral Calculus. London, 1842.
Dubamel, Cours d'Analyse. Paris, 1847.
Euler, Institutiones Calculi Integralis. Petersburgh, 1792.
G(regory's Examples on the Diff. and Int. Cal. Cambridge.
Hirsch, Integraltafeln. Berlin, 1810.
Lacroix, Calcul Diff. et Integral. Paris, 1797.
Lagrange, Leqons sur le Calcul de Fonctions. Paris, 1806.
Landen's Residual Analysis. London, 1758.
Legendre, Exercices du Calcul Integral. Paris, 1816.
- TraitS de Fonctions Elliptiques, 1825-8.
Littrow, Anleitung zur hdheren Mathematik. Vienna, 1836.
Mending's Tables of Integrals.
Ohm (Martin), System der Mathematik, 1833-51.
Raabe, Die Differential und Integral Rechnung mit Functionen Mehrerer
Variabeln.
Schldmlicb, Handbuch der Differenzial Rechnung, 1847.
Taylor, AMethodus Incrementorum. London, 1715.
INTEGRAL CALCULUS.
SECTION III.
DEFINITIONS.-GENERAL PRINCIPLES OF INTEGRATION.
11. QUANTITIES are said to be functions of one another, if
their values depend in any manner on each other. The
letters F,f, (, &c., prefixed to quantities, are used to denote
functions of them. A function of several quantities is expressed by writing the letters F, f, &c., before them all separated by commas.
12. A variable is a symbol of quantity to which different
values may be assigned.
13. An independent variable is a symbol of quantity, on
the value of which the value of a function of it is considered
dependent.
14. A limit is the exact value which a function approaches
nearest, as the variables on which it depends approach assigned
values.
15. The limit of a finite continuous function of several
quantities is the same function of their limits, or if yp, y2, Y... be the limits of y, y, y3... respectively,
limit of f (y&, y, y...) = f (y, y, y...)...... (1),
where f means " any finite continuous function of."
A continuous function is one such that the series of operations denoted by it when performed on more and more nearly
equal quantities, produce more and more nearly equal results;.*. f (yU, y, y...) -/f(yr y., y,...)...... (2),
is smaller, as y, y2, y, &c., are more and more nearly equal
to y,, yY, y,, &c., respectively. Therefore, the limit of the
finite quantity (2) is zero, or
PRINCIPLES OF INTEGRATION. 15
limit of f {(y,, y 3,...) - f (y1, Y y3...)} = 0,
from which equation (1) immediately follows.
16. The quadrature of a finite continuous function of one
variable having a limited range of values is the sum of products of successive values of that function, each multiplied
by the differences between the corresponding value of the
independent variable and the next preceding or succeeding
value.
17. The integral of such a function is the limit which its
quadrature has when the differences of the independent variable approach zero, and their number approaches infinity.
18. Let fx denote a finite continuous function of x, and
let b, and bz be two constant assigned values of x. Also, let
c,, x,2 x3.. V,, be any successive intermediate variable values
of x. Then the quadrature of f, is by the definition, either
f x~,- b,) + f X2(,(-X1 ) + f Xa- 3, )+... +/6,(b, -,,),
or fb (x - b1) +fx, (x - ) +/fx, ( - ) + - x. + +f,, (b. - x,,).
The integral of the function is the limit which these series
approach when the differences x1 - b,, x - x,, &c.,.approach
zero, and their number infinity.
19. In Art. 7, let x be the abscissa, measured from B
along BC of any point in the curve BA, and let fx denote the
corresponding abscissa. Then it is clear that the differences
X,-- b 2, x,--x,, &c., denote the breadth of the rectangles
drawn in the figure, and fv,, fx2, &c., the corresponding
altitudes. Hence, the several terms in the foregoing series
denote the areas of those rectangles, and their sum is an
approximation to the curvilinear area ABC, whence the term
quadrature is derived, since that quantity expresses approximately the number of square units (square feet, square yards,
&c.) contained in ABC. Also, the integral is the exact area
ABC; for the magnitude of this area is between the magnitudes of the inscribed and circumscribed figures. But the
difference between the two latter magnitudes has the limit
zero. A fortiori, the curvilinear area differs from either of
them, by a magnitude which has the limit zero.
INTEGRAL CALCULUS.
As the figure last referred to is drawn, the initial values of
x and of fx are both a
supposed to be zero. If,
however, they be finite
positive quantities, the
integral represents an
area such as abcd, where
o is the origin from which
the abscissae are drawn,
and
oc = b, bc =fb1,
od = b, and od= fb6,. c a
20. Both expressions for the quadrature in Article 18 have
the same limit, if fx have only one finite value for each
value of x from b, to by, for then they differ by the quantity
(fX -Af ) (X - b- ) + (f2 -f ) ( - X1) +
(fx,- f x,) (x, - x) +... + ( fb2 - f n) (b - x,,).
Let x be the greatest of the successive differences of x
in the preceding quantity, which is therefore less than
(fi -fb)6 A + (fx2 - f.) A x +.. + (fb, - fx,) x,
which expression is equal to (fb. - fb6) x. This, therefore, is the difference between the two quadratures; but if
fb, and fb, be finite, fb2 -fb, is finite; A is zero in the
limit. Therefore, the difference between the two quadratures is zero in the limit, i. e., they have the same limit.
21. The preceding article is exactly illustrated by the
Lemma iii. of Newton's Principia, which is as follows (supposing all the parallelograms spoken of in the original to
be rectangles):In the plane figure bounded by the curve AF and straight
lines AA', AF, at right angles to each other, are inscribed
any number of rectangles AB', BC', CD'... on unequal
bases AB, BC, CD..., and the rectangles AB", BC", CD"... are completed. If the breadth of these rectangles be
diminished, and their number increased indefinitely, the in
PRINCIPLES OF INTEGRATION.
17
scribed figure AKB'L C' MD'NE'E, and the circumscribed
figure A A' B" B' C" C' D" D' E" E F are ultimately equal
For let Af be equal to
the greatest breadth A^ 'I
of the rectangles, and K -~,
complete the rectangle I -- -
Af', then this parallel- EN
ogram will be greater M D--Bthan the difference between the inscribed I N El
and the circumscribed I
figures. But when
its breadth is dimi- __
nished, it will be less A 13 C i, E F
than any assignable
quantity, and, therefore, a fortiori, the difference between
the inscribed and circumscribed figures will be less than
any assignable quantity, and, therefore, they are ultimately
equal.
22. When fx continually increases or continually decreases,
as x increases, the value of the integral is between those of its
quadratures. First, let fx continually increase as x increases, then the integral is less than the first quadrature,
Art. 18; for let x' and x" be any two successive values of
x, then one of the terms of this quadrature is fx"(x" - x').
Now, take a value x, between x' and x", then the term
in question is replaced by
fx, (, - x') + f" (W" - x),
which is less than the term just mentioned by
( fe" - fx,) (a1 - X'),
a quantity which is positive, since fe" is always greater
than fx'; therefore, the effect of increasing the number of
terms is to diminish the quadrature. But as the number
of terms is increased, the value of the integral is more and
more nearly approached; therefore, the integral is less than
the first quadrature.
Similarly may it be shown that the integral is greater than
the second quadrature.
The same reasoning may be applied when the function fx
continually decreases as x increases; therefore, in either case,
the integral has a value between those of its quadratures.
18 INTEGRAL CALCULUS.
~3. The symbol of integration is f, which derives its
form from the initial letter of the word Summa, or Jfm.
The integral of a function fx of a variable x is written
ffx. dx; where the limit of the difference between two successive values of x is represented by dx, which is, therefore,
differential, or diminished without limit; and fx. dx is the
general form of the limit of any term of the series in Art. 7,
and is also differential.
ý4. The limits of an integral are the two constant assigned
values of the independent variable b, and b2, in Art. 7. The
greater and less of these values are frequently designated
the superior and inferior limit respectively.
25. When the limits of an integral are expressed, or
defined, it is said to be definite; when they are not defined,
indefinite. In the first case, the integral is said to be taken
between limits. The usual way of expressing this symbolically
is, by writing the superior limit above, and the inferior below,
the symbol of integration. Thus, fx. dx is the integral
of fx, between limits b, and b6.
26. The value of the integral is independent of the differences of the independent variable in the quadrature. For the
limit of the quadrature is, by Art 14, an exact quantity, therefore it cannot depend on the values x, X, x.1... x,, nor their
differences, which may be altered arbitrarily. Also, it is
evident that the integral does not involve any other values of
x, except b, and b,.
COROLLARY. Hence jfxdx fzdz, where z is
any other quantity than x.
07. The sum of definite integrals, the inferior limit of each
being the superior limit of the next. If the series in Art. 18
were continued to the right, to the term in which x = b,
the limit of this additional part of the series would, by the
preceding definitions, be J fx. dx. Also, the limit of the
whole series, including the additional part, would be
Sbfx. dx. But this whole series is the sum of that written
in Art. 18, +- the supposed additional part. Hence,
PRINCIPLES OF INTEGRATION.
19
' dx =/ x dx +fjf4, dx......... (1)
Similarly,
=, pb. p-,Sfx = d=f fx. dx + f.dx +... +
bi b ra-l n-a
f.d + x. dx.
28. An Integral between limits is the difference between two
values of the same fanction. By Art. 26, fix dx is independent of all the values of x, except b3 and b,. Therefore
this integral may be put equal to F (b,, b,), some function
which contains no value of x except b1 and b. Similarly, if
the form of this function be general, that is, capable of representing the integral for all values of the limits, / jfxidx=
F (b,, b,). Hence, from (1) Art. 27, transposing,
f'yx. ddx = F (b,, b,) - F (6,,,);
but fx. dx involves no other value of x than b, and b,.
Therefore bi disappears from the last equation, which, consequently, may be written
Sfx. dx = Fb3 -F,;
COROLLARY, /fx dix =dx - >fx da.
29. By Article 26, the value of the integral is independent
of the differences x - b, x2 - 1, &c., in Art. 18. We may
therefore suppose those differences all =b x, so that
(n + 1) $x = b2 - b. Then, by Art. 28,
limit of (f +f/ +fx +... + f, +fb ) x = Fb - Fb.
The number of terms in the parenthesis is n + 1. Now
suppose, first, that the fx is always positive; and let fd'
be its greatest, fx" its least value between the limits;
INTEGRAL CALCULUS.
then fx' is greater and fx" less than any other of the
terms in the parenthesis. Hence (n + 1) jx' is greater,
and (n + 1) fx" is less than their sum;.. (n+ 1)fx'.x>Fb-Fb,; (n + 1)fx"/ x<Fb2-Fb;
or, putting (n + 1) 3x =h; kf ' > Fb - Fb;
h fx" < F b' - F b6.
There must therefore be one or more values of x between x,
and b2, for which kfx = F b, - F b~. But this intermediate
value of x must also be between b, and by, since xa may be
taken as near b6 as we please. Therefore the intermediate
value in question may be expressed by b, + Oh, where 0 is
some positive proper fraction. Hence, since we have supposed b, = b, + h, we have the formula
lif(b, + O) = F (b + h) -F rb = b'1+7 f. dx
The same conclusion would be arrived at if fx were supposed to be always negative. Hence the formula is true
when fx is either always positive or always negative between the limits b6 and b + h.
30. The following is a geometrical illustration of the
formula h f (b, + 0 h) =
Sbl+ h.......
fx dx.
Let f x represent, as in
Art. 19, the ordinates of the
curve ab, and x its abscissa,
measured from o along
od; oc = b, od = b, + h;.. cd==h. Also bc =fb1; b
ad = f (b, + h). Then the oc e
area abcd = bh fxdx. Now the formula asserts that
bi
between bc and ad there is some intermediate ordinate represented byfe in the figure, and by f (b, + Oh) in the formula,
such that fe x cd = area a(bcd, a proposition which, from
geometrical considerations, is evidently true.
PRINCIPLES OF INTEGRATION.
31. A Function is the di'ferential coefficient of its Integral
Dividing by h, the result in Article 29,
F (b, + h) -- F
f (b1 + )h) =
Taking the limit of both sides of this equation, when h has
the limit zero,
fb, = differential coefficient of F b,
by the definition of a differential coefficient. Hence is seen
that INTEGRATION IS THE OPERATION INVERSE OF DIFFERENTIATION.
32. The integral of the sum of several functions between
given limits = the sum of the integrals of the several functions between the same limits. Let the several functions be
fAfA,... *f,,,
Sxfdxd =limit of (fx +fi9X/ + fX3 +...f b2) x
f x d =d limit of (f,2x, +f22 +...fb') lx,''f.f dx = limit of (f,, +f, x2 +f +...f b,)$x
Adding, fx dx + fx dx +... + b f, dx =
limit of {(f X+ +f2x, +... +f"X,) +(f x+/f2 x+.... +fnX)+
&c. + (fixn +fAx, +... +fx,)} ax =
f ix' +jf2 +... +f/x) dx.
33. A constant multiplied by the integral of function between
given limits = the integral of the function multiplied by the
constant between the same limits. Let c be the constant. Then
INTEGRAL CALCULUS.
c / fx dx = c limit (fx + fx +fx +... + fb) x
= (by Art. 15) limit of (cfx1 + cfx + cfx3 +... cfb~ ) ~x
=/ cfx. dx.
34. To show that J b y dx + u dy = bc, - b c, Iif
y be a function of u, and have the values c,, c, when u has the
values bb, b2, respectively., 3/'2, Y2... y,, being successive
values, the function y and u, u2... u, of u, we have, by
Art. 18,
7j2
y du = limit of cI (u1 - b) + Y, (u2 - u1) +
Y, (u3 - i) +... + (b - u,,)}
Cu dy = limit of {uI (y, - c) +
U (U2 - Si) +. + ( - ) + +, ( (c, - Y))}.
By adding together the quantities in the { }, it will be found
that all in each line except one appear in the other line with
contrary signs. So that the sum in question is reduced to
b c, - -b c1. Hence
jby du + udf y = b,2, - b2.
35. The conclusion of Art. 34 may be arrived at from geometrical considerations, as follows:
Let AB be a curve re- y
ferred to, Ox, Oy as
axes of co-ordinates. Let Fr a A
OC = b, OD -= b.
Then the area ABCD=
yfdx.
In the same way, if I
OE = cl, OF= c2, the c
area ABEF = y. o x
PRINCIPLES OF INTEGRATION. 23
Therefore "y dx +j c xdy = figure AFEBCD =
rectangle AO - rectangle BO = b, c, - bi ci
36. To determine fdx. In the first equation, Art. 29, it
is not necessary that fx should be variable. Let it = 1.
Then limit of (Sx + Sx +... + 4x) f dx.
But, evidently, the left-hand side of this equation -= b -,,1.6. b- - b d, a.
37. If x and y be functions of each other, so that
fx fdx = f Y dy (1), and m= b when y= c,
then fx dx =; y dy.
For let (Art. 28) the first of these integrals = F - F b,
and the second = h y - ( c. Then
Fx - Fb= y-- c.
Let x become x + x when y becomes y +,y. Then
F (x + x) - F b = D (y + y) -- (c.
Subtracting the last equation from this,
or F(x+ x)-F 4(+(yY)-Dy W,
or.y.. (-)"
Now this equation is true, however small ýx and Vy may be;
therefore, the limits of both sides (corresponding to the limit
zero of ýx and Sy) are equal; or, by Art. 17,
fx = Y, or f. =1 y............ (3),
whence, fe div = ýy dy.
INTEGRAL CALCULUS.
38. To prove that if fx dx = gy dy, and x be equatl to
b2 and b, when y is equal to c. and c, respectively, then
fI dx = C2y dy.
fb2 f CJC2 +f C
For let Jb 1fx de J'C y dy Cf C2 y dy,
then, by the last proposition, fw = y + p,y.
But by the hypothesis fx = qy,. l)y = 0,.* qcIy dy = 0,
for this last integral is the limit of the sum of a series
of which the terms are all absolutely zero;.j f. df = 9cY. dy=f f dy; by ().
I c fe j; U d, ),
39. From the preceding article follow many important relations among definite integrals. For instance, let y+ a =x;
then c, 2+ a =b2, c, + a = b,, dy = d;. fx = qy
f(y - a), and the formula becomes
( - b C2 152 - a,
(y- a) dy = cayd= J fVdmx.
Now in the first of these integrals we rmay, by the Corollary,
Art. 26, write y for x. Therefore
fl2f (Xf- a) d1 ffd........=- O..... (I.)
Similarly,
f (x + a) dx = +a fxdxv................ (II.)
b2bf(b - x) d f = b2s............... (III.)
Putting y - a = x and b. -y = x successively.
PRINCIPLES OF INTEGRATION.
Q5
Putting y=- x; "fxdx == f(-x)dx... (IV.)
And generally, if x =,y, whence y = {x, dx = 'ydy,
Sfxdx f()x xdxI......... (V.)
40. Indefinite Integration. We have shown that if
function can be integrated between any limits a and)
its independent variable, the integral is of the form F (a) -
F(b). There is a large class of functions which cannot be
thus integrated between all limits, or of which the general
integral cannot be found. The first part, however, of the
science of integration, is confined to the investigation of
general integrals. Our object is, therefore, to find the form
of the function F, which represents the result of the integration of the function f. It is not necessary for this purpose to find F a - F b, but, simply, F x, from which F a - F b
may be found by substituting a and b successively for x, and
subtracting. In the following chapter, therefore, Fx alone is
required.
COROLLARY. It follows that the formula of Art 34 may
be written
fy duz +Jfu dy = zny, orfy f d d= iy -Zfu dy.
41. Diferentiation of Integrals.
From (a) and (/3), Art. 37, it follows that
d fx dx =fx = fxdx; or, writing a for x,
d / a
-d fa da = fa; or, by corollary (Art. ý6),
(a fxdx = fa. Also,
(I f xdx = - fxd = - fb.
From the first of these equations, it appears that the
differentiation of an integral may be performed under the
sign of integration.
INTEGRAL CALCULUS.
SECTION IV.
FUNDAMENTAL INTEGRALS.
4F. To integrate acdx where a is a positive finite quantity.
By Art. 15, putting x, = b, + -x, x = b, +- 2x, &c.,
x. = b, + nJ, b2 = b1 + (n + 1)+,
a' dx
= limit of (al + x + a` + +2'x +... ab, + (t + 1)X') x
=limit of ab,+3x (1 + ax - a2 - +... a') ýx
Slimit of ai +x a x a( - + 1) _1
= limit of as + ax n
ax - I
= limit of (a,+ -+x a,+x).........(1.)
a6x - 1
Now the quadrature of which the limit is here to be taken
is finite, since all the quantities are finite. By Art. 2,, the
integral of such a function as a" has a value between those of
the two quadratures, from which it may be obtained. But
the quadratures evidently may here be finite quantities with
the same sign. Therefore, the integral between them is not
zero, nor infinite.
It follows that in (1) the limit of a- is some exact
3 ^x - 1
function of a. Call it A. Then taking the limit of (1)
a dx = A (ab2 - aC). Also, fax dx = Aa.
If A be such a function of a that A = 1 when a has some
vialue c, /f., -
FUNDAMENTAL INTEGRALS. 27
Also, faxdx =felogs adx - 1 flg, - xdd(log, a. x)
10g a al
0 a1
- log,anx - ___
log, a log$ a
dx
43. To integrate -. Let x = Ec, and when y = c, c1, c2,
let x = 6, bp, b2, respectively. Then
X - b = I,r'
_ b=cY-f
but x - b = dx, and e c- Cy d,
by Art. 36, and the last article respectively. Hence by
Art. 37,
de
dx = e dy; dx = dy.
Therefore, by Art. 38,
b1 dav c,
bdy = c. - cl, 10g, b, - log, 6,,
since if x =, y = log. The indefinite integral is
J de
d = log, x.
44. To integrate,t, where a is a finite constant and z
variable.
Let y - a+, or log, y = (a + 1) log x, a having any
real value except - 1; when y = c, or c,, or c2, let x= 6, or
b,, or b6, respectively. Then
log. y - log, c = (a + 1)(log - log, ).
By the last article, log, y - log, c = dy
and (a + 1) (log x - log, b) = (a + i)/
c52
ý8
INTEGRAL CALCULUS.
Then by Art. 3'7, (a + 1) d
y w
x
o 1 c, I
d(+)x 1 X
Hence f2. dx = 1 dy C (c - cC), a +, a +
1 /a+1
= (C9a+1 - c a+). Also, xadx = xa+
a+1 " a+1
45. JOHN BERNOUILLI'S series. By repeated integration
by parts, and Arts. 37 and 44, we have,
fox x dX
Xdx = Xx- x dx
fo f dz
x2 dX rx x2 d2 X
= X - - dx
ý2 dx fo dv
x2 dX X3 d2X fx x3 d3X
SXx- - + - I dx
2 dx 2.3 dx2 o I.3 dxdd
= &c.
x2 dX x3 d2X x" d" X
-X dx +.. dx'
2 dx 2.3 dx.o 2.3.4...n dx"
On the second side of this equation all the quantities are
taken between the required limits x and 0; since each is zero
dX d2X
for the latter limit; X, ' d... being supposed to be
always finite.
If the last term of this series become zero when n is
sufficiently increased, we have
fzx x dX x d2x
dx = Xx - - +... ad infinitum.
- o 2 dx 2.3 dx
By Art. 29,
x x" d" X
xf(Ox) = fxdx. Put 2.3.4...n dx" - fx,
FUNDAMENTAL INTEGRALS.
Hence, a criterion that the last series may be continued
ad infinitum, is, that f(Ox) become zero when n is suffix9 d" X
ciently large, or that then - = 0 for all values
2. 3. 4... n d"
of x between the limits x and 0.
46. e is the base of the Napierian logarithms. By Art. 42,
fE' dx = e............................. (1.).f. J dx = -.fe-xd (- x) = -...... (2.)
Therefore, in Bernouilli's series (Art. 45), if
X=e-, dx _- d2 x _.
dx ' dx2 '
Hence the series becomes
E- d= + = I.3 ~+2+ -3 -}.+ C
For all values of x in this series the criterion of Art. 45
is satisfied, so that the series may be continued ad infinitum.
The first side of the equation by (2) is equal to - e-" taken
between limits 0 and x, or = - (e-x -1)
-(e-x-1)= x+ + +.... eDividing by e-", and transferring one term to the second
side of the equation
e" = + + +...
In this equation put x = 1. Then
1 1
e= + 1 + 2 + + +'''
Therefore, e is the base of the Napierian logarithms.
INTEGRAL CALCULUS.
47. To integrate sin xdx. sin xex = limit of
{sin (b, + x) + sin.(, + S 5 ) +... + sin (b, + n + 1..x)} x,
where b2 = b, + (n + 1)) X.
By a known trigonometrical formula,
cos (A - B) - cos (A + B) = 2 sin A sin B.
Therefore, putting B =. x
2sin(b, + 3x) sin 4 x = cos (b, + x) - cos (b, + )
2 sin (b, + 2 x) sin 4 x = (cos b, + I- z) - (cos b, + x)
Adding these equations,
2 sin. J {sin (b, + x) + sin (b, + a x) +...
+ sin (b, + n )+ 1 x)}
= cos (b, + 4 X ~) - cos (b, +? + 4 )
= cos (4, + ) ') - cos (b, + 4x);
pb2.*. / sin xdx = limit of
cos(b, + ) - cos(^ 4-m)
sin 4 Sx 2
Assuming the demonstration given in the subsequent section on Rectification of Curves, that the limit of I 2x
sin i 'x -= 1 when ' x has the limit 0, we have,
2sin xdx = cos b - cos b. Also, sin xdx = - cos x.
dbl f
FUNDAMENTAL INTEGRALS. 81
48. To integrate cos xdx. /j cos xdx = limit of
[cos (b, + ýx) + cos (b, + Q 3m) +... cos(b, + n + 1 w)} IXT,
where b, = b, + (i + 1) ýX.
By the trigonometrical formula
sin (A + B) - sin (A + B) = 2 cos A sin B;
we have, putting B = ýx,
0,cos (b, + ým) sin -1 Lm= sin (b, + ýx) - sin (b + - 3z)
2cos (b, + 2x)sin m= sin (6,+ Lx) - sin (b + Az)
sin.1 {cos (b, + ýx) + cos (b, + ý ýx) +
cos(b, + 3 3) +... + cos (6, + n + 1 x)}
-sin (b,+ 1, ) + sin (b, + 1 '),. C. cos id = limit of
sin (b, + x) - sin (6, + x) i
2.sin S x 2 ýx = sin bz - sin bp,
sin 2 Ix
putting limit of I x + sin x = 1, as in the last article.
Also, fcos vdx = sin x.
This integral may be obtained immediately from the proceding article, for
Jos d v = - sin ( -a d - )=
(by the last article) cos - = sin x.
39 INTEGRAL CALCULUS.
sin xde cos xdx
40. "To integrate and. Since
cos2x sin2 x
J' sin xad = - cos x,.. d cos = - sin xdx,. xdx cos by Art. 44,
cos dx Cos x Cos
50. Similarly,, -cosd x - sin _
sinl2 ' sin
51. To integrate (1 + tan x) dx.
sin" x dx sin x d cos x
tan2 xdx = -- y du,
CoS2 X COS 2,
d cos x
if y = sin x and du =ds
Cos 2.. by the last article u=, also dy = cos x dx.
cos x
Now, by Art. 40, J/du = yu -J'udy,
~ tan2 dx sin x 'ccos xdx tan x --.
cos d Cos -
Therefore, /(1 + tanx) dx = x + ftan2 zdx = tan w.
5.. Similarly, J(1 + cotan'2 ) dx will be found to be
- cotan x,
orf/(1 + cotan2 x) dx =
S 1 + tan -- ) d --x) = - tan --x
(as has been just proved) =--cotan x.
FUNDAMENTAL INTEGRALS.
33
dx
53. To integrate -. If a be not zero,
Now, dx = d - a),.1 a f= logI ( x - a) (Art. 43),
-a I X+
_ = { log8 (x -- a)- log8 (x + a)}
Now, dx = d (x - a),
a = /-d-- log, (x - a) (Art. 43),
f X c d( - a) C
a 2 f~- =-,log, (X - a) - logs (X + a)J
log
I x $- a
-2a gx + a
If x be less than a, the logarithm just found is the logarithm of a negative quantity; and is, therefore, impossible.
In order to express the integral in a possible form in this
case, put
rddx d 1 f dx r dx
/xV - ca2J -x2 ~a a V~ x
S1 a-x
{- log (c, - X) + log (a + w)} =I log
2a Qa a + x
dx
54. To integrate (x
Let dy = dx + ---................()
(X2 ~4 a )2
Now f ((xdx
N2ow = 'f xq 4 a+)-' d~x- - a
S(X9 t - t N
(x?~I a?) (Art. 44)..*. y = (X2 4 a2).
c3
INTEGRAL CALCULUS.
Also from (1),
dy dx, a) dy ldx
Hence, / ---dx2) = loog, =og, {(,V + (x2_ ~ ) }.
dz
55. To integrate--- --- where, in order that the
denominator may be possible, da is greater than x1, if x- be
affected by the negative sign. In Art. 54, write - for a,
t 1
- for x, and, therefore, -. d for dx.
x x
m r ~ -2 w-' d dx
Then =- -2X - a dx
a - (a x(- x~ )x
=logs {x-1 + (x-2 -- a-) }= log, (a
ax
S/ dx 1 ax
-_____ - - log. (x9 4-~ a a a, + ( 4-~ 2)
(since the logarithm of any quantity = - the logarithm of
its reciprocal),
1 o x 1
= -log +- - + log a,
a - a + (a'~ +- ) x)' a
of which expression the last term - log, a may be omitted,
as it disappears when the integral is taken between limits,
dx
56. To integrate: where a > x.
(a2 - )
Let dx = cosydy. Then (Art. 48),
x = sin y, (1 - x2) = cosy,
FUNDAMENTALI INTEGRALS.
de cosy dy
dx f os = y = sin>- x. Hence,
(1 a d)2 'Cosy
d -
/ dxr dx'
- --- a x =d '
in -- -cos-1 - = os- - if - be included in
a 2 a a 2
the value of the integral at its inferior limit.
57. To integrate - Let da = sin y dy. The
x (X2 - x i.
a
integral of the first side of this equation is - -, and of the
second - cos y;. we may therefore put - = cos y. Hence
(Xa - ) d x d y sin y
= sin y, and=
x x.(- a) x' sin y
Sy 1 _ a s e a
--1--.
a x ( a a a
58. To integrate dx
( ('ax - (a a )
a - --
C os-'1 (Art. 56) = versin-I
a.
dx -a
59. To integrate. Let - = tan y,
+a
dx = a (1 + tan' y) dy (Art. 51),: da (I + tan2 y) dy - 1
Ct (1. + tan2 y)
S- tan. 1
a a
36 INTEGRAL CALCULUS.
Collecting the results of this Chapter, we have the following
TABLE OF FUNDAMENTAL INTEGRALS.
aTv ARTICLE
Ja d =................................. 42
log, a
O1
Sdx = 7+1 except n = - 1 when............. 44
J = log,..X............s.................s............. 43
sin a.d = - cos x.................................... 47
J cos x. dx = sin x............................. 48
i x dx -....................................... 4 9
cos- x COS X,kfcos x 1
sin x Ix
dx =,......,....,.,......,.,........,50
J (1 + tanx)2 d = tan x......................... 51
( + cotan2 x) dx = - cotan x.................. 52
dx 1 lo x - (x a >a)
f X _C// a "X + a~
1 a-x
S logag, (x < a)............... 53
2a a+x
dx = log, { (x2 a) }...............54
f ( 2--- a 2-4 2)j
Sd 1 x
= logs 5 5
x (a2 - ) a (c2 -...........
dxl X X
sinx, or-cosST -............s
a a
FUNDAMENTAL INTEGRALS.
37
a 1 Ca 1 X
a,I - () X a, a,
Sversin 1 x
f dx 1 ta -1 59
/ fa tan- 1 -.......................o............ 59
Wa" +l_" a n a
60. The foregoing integrals are all found in terms of logarithmic, exponential, and circular functions. Tables may be
obtained which contain numerical values of these functions
computed to any required degree of accuracy. Therefore the
values of these integrals may be completely determined.
Similarly, other integrals which can be reduced to any of the
forms in the preceding list, may be completely determined.
61. The operations of integration consist chiefly in reducing
integrals to these fundamental forms. In many cases, however, this reduction cannot be effected by known methods.
Where it is impracticable, resort is had to methods of expressing integrals in terms of convergent algebraical series, or in
terms of elliptic and other functions not contained in the
preceding list, but which have been partially tabulated.
6o. For the present, however, attention will be confined to
those integrals which can be reduced to the forms investigated
above. The methods of effecting this reduction may be
classified as follows:
1. Integration by Algebraical Transformation.
2. Integration by Parts.
3. Integration by Formule of Reduction.
4. Integration by Rational Fractions.
5. Integration by Rationalization.
Of each of these five methods a brief account will be
given in the following sections.
INTEGRAL CALCULUS.
SECTION V.
INTEGRATION BY ALGEBRAICAL TRANSFORMATION.
63. THms method, of which instances occurred in Arts. 54,
56, &c., consists in finding for the expressions to be integrated algebraical equivalents which are of the forms of one
of the fundamental integrals, or are the sum of quantities
having any of those forms. The requisite transformation
is effected by substitutions and other processes, for which
no general rule can be given. It is only by continual
practice and experience of the effect of various transformations that facility in the successful application of this method
of integration can be attained. One or two examples are
appended, but for an adequate knowledge of the subject, the
student must be referred to larger collections of examples of
the Integral Calculus.
64. Every polynomial of the form (a + b6 + c - +...) dx,
may be integrated in finite terms when n is a positive integer,
and the number of constants a, b, c, &c., finite. For the polynomial may be raised to the power n; the result is the sum
of a finite number of terms involving only integral powers
of x, and each term may be integrated separately.
65. For example f(a +- b)' dx =J/f(a?2 + ab x+- b") )dx
2 3
66. If the function to be integrated can be expressed
as the product of two quantities, Fx, and dFx, or more generally (Fx)"', and dFx, it may be always integrated. For if
Fx be put = y, the expression takes the form y" dy, of which
the integral (Art. 44) is -.
67. For example, (a + bx + cx2) (6 + 2 c) dx becomes,
if a + b x + c x' = y, J' = y = (a + bx + c).
ALGEBRAICAL TRANSFORNMATION. 39
S68. Again, (logsg ) - =o (log = d (log, x)
(logs X)11+l
n+l
dx dfje+ - d (d)
69.
t + f-x + +
+ d + =-x)
fd + - log(1 + E-).
'70O. All the preceding formulk for integrals of functions of
x may be extended to like functions of a + bx, by putting
I
a + bx = X,.. b dx = dX, and dx = dX.
In this manner it will be found that
aa+ bx dx= t+
b lo., a
dx dx = I 10g, (a + b X)
a+ +bx b
__ 1 (a + 6 m)n
(a + b z)" dX- ( n+1 except n=-1
1 n+
sin (ac ( + )x) dx = os(a+ )
cos( )d i (a + b d = Si (a + X)
1 + tan2 (a + b) } dx = tan (a + b)
J i + cotan2(a + bh)} d = -d cotan (a + bx).
71. A similar extension of formula for functions of
a -~ X, to like functions of a + bx + czx2, where a, 6, and
40
INTEGRAL CALCULUS
c are positive or negative, may be effected by the following
transformation:
a+ b+cx -=c+ { 2 = ab c(A y2),
a b2
if - =- A, where A may be positive or negative, and
b
- x = y,.'. dx = dy.
Hence it will be found that
/ dx 1 dy
fa + bx + c ' c ~f + A
1 1 y-- (--A)a
- log y - Art. 53,
-L
c a (- A)' y + (- A)
(A negative)
11 y
= tan -1 L- (A positive), Art. 59.
cA A
f/+ d 1 I JAdy
J(a + b x + c) cx (A + y")
=, log, {y + (y2 + A) } (c positive,
A positive or negative), Art. 54.
1 y
= ( sin l (A and c negative),
Art 56,
(impossible if A be positive and c negative).
INTEGRATION BY PARTS.
41
SECTION VI.
INTEGRATION BY PARTS.
72. A FORMULA has been given, in Art. 40, of which very
extensive use is made in integration, and of which applications have been already given in Art. 45 and 51. This formula, called the formula of integration by parts, is
f udv = uv -- 'vdu.
Any differential function of one independent variable may
be put in the form udv. If, then,fvdu can be found,fudv
can also be determined by the preceding formula.
73. To integrate x log, xdx. Let log, x = u, whence
d x
== du (Art. 43). Also let xdx = dv, whence xw = v,
(Art. 4.),
'f v logs xdx =fu d v = uv - fvdu
1 o 1,dx
=- logX- -x'2 2 X
= x log,. - _ x4.
74. To integrate x dx. Let x dx = d v. Then Ex=,
Art. 41. Also, let x = u; d. = du.
JxFx dl = fudv uv -J' f Elu = xEs -fas d
SaX -
x xdx 2 x dx
75. To integrate - X Let dv = (1 d--x
(()(1 - X.)
(1 - x),.. (Art. 44) v = Also let u= x.
(1 - ')'- 1 - X
INTEGRAL CALCULUS.
The formula gives
d d 1 m ' dx
'I1-) -2 1- # - 1 -#
1 _ 1 d
1 - 1 r-1
Sx " V 1
+- log
2 1-X" 4 1 + 1
76. To integrate dx (a - dv). Since
a;-.) 9- a2), by Art. 41.
a (a' -
xdw
Therefore, d(a2 - 2) =
Hence, integrating by parts,
~/d~(o?-C) 9Y. + X2 l ~
fda (aI - = r (i- +
consequently, transferring to the first side of the equation
the last member of the second side, we have
1 1
dx (a - x) = x (a - v) + a sin'
77. To integrate x cos x d. Putting fcos edx = sin z,
we have f2cosxdx = x sinx -fsin dx
= x s x + COS.
TNTEGRBATION BY PARTS. 43
78. To integrate EX cosxdx. Performing the operation of
integration by parts twice,
fe' cos xdx = e" cos x +fcxe sin xdx
= eC COS X + e" sin x -fEx cos xdx.
Transposing and dividing both sides of the resulting equation by 2,
Jex cos XdX = e (cos x + sin x).
INTEGRAL CALCULUS.
SECTION VII.
FORMUL~E OF REDUCTION.
79. By Formule of Reduction, integrals involving powers
of functions are expressed by integrals involving higher or
lower powers of the same functions. These formule are
obtained by the principles of integration by parts and algebraical transformation.
80. For instance, the integral of x' cos x may be made to
depend on a function of '"-'; the latter, similarly, on a
function of x"-2, and so on continually. If m be a positive
integer, and the process be continued a sufficient number of
times, the last integral is that of cos x or sin x, which have
been found in Art. 47 and 48.
Integrating by parts,
fx"' cos x = x"' sin x - mj.. -' sinx d x
= X"n sin x + mx "-' cos x - m. m- 1. fx1-2 cos xdx
= x" sin x + mx'"' cos x - m.m - 1 "-2 sinx -
m. m- 1. m - 2. "3 cos X + &c.
the positive and negative signs succeeding in pairs.
For instance, let m = 4
jx cos xdx = x' sin x - 4 fx" sin xdx
= x4 sin x + 4. x3 cos x - 3.4 fx2 cos xdx
= x sinx, + 4. x3 cos x - 3.4 z2 sin x + 3.4. 2.fx sin x dx
= sinx + 4 x3 cos x - 3.4. x sinx -
3.4.2. xcos x+ 3.4.2.1sinx.
81. The preceding integral is an instance of a general
formula which is an extension of John Bernouilli's series.
By the same method as that by which Bernouilli's series was
obtained (Art. 45), we have, if P and Q be functions of x,
FORMULXE OF REDUCTION.
and Q', Q", Q"'... successive differential coefficients of Q
with respect to x, and
P1 =fPdx, P, = JP, dz, P, =fP, dx, &c.
fPQ dx = QP, -f Q'P, dx
= QP - Q'P, +f "P dx = &c.
QP, - Q'P, + Q"P3 - Q"'P4 + Q""P, -... fQ") P,, dx.
8,. To integrate x" ex, n being a positive integer. Here
Q = t, Q1 = ne@-1, Q" = n. n - I. "-2,
Q"' = n n - 1. n - 2. x.-3 &c.,
Q(') = n.n - 1. 1 C,P = P e, P, =eX, &c.
Therefore,
J.nE ex d x = 7" - 1 n z"t- ex + n. n - 1. -2 e"~...
n.n- 1... 2.1.fe 'dx
= e" (X' - n x-1 + n. n - 1. x" -...
Sn.n- 1... 2.1).
The formula of the last article but one is inapplicable,
except where the successive integrals PV, P2, P3... are simple
quantities, and Q(") such that fQ"P,dx may be found. This
will not generally be the case for functions involving fractional indices. Such functions may, however, be frequently
reduced by combining integration by parts with algebraical
transformation, as in the following example:83. To integrate (a2 - x2')2dx, n being an odd integer.
In the formula for integration by parts
fudv = uv - fvdu, let (a2 - 2)% = u.
Then - nx (a- - )2 dx - du; dv = dx.
f(a' - d) d = (0, - x2) X + n (a2 _ X2) 2 - X.. (1)
INTEGRAL CALCULUS.
Now,
(a2- x) 'I V- (-a2 ) (-a2 (a + a" (a' x)
Integrating this equation, fJ((2 - x) dx =
9a/(, - d) -' dX - n.f( - 2-1 x2dx... (2.)
Adding (1) and (2), and dividing both sides of the resulting equation by n + 1,
n 1 f(a-- +x2) d x.
By this formula of reduction, the integral is made to
depend ultimately on f(a- - )-dx, which has been found
in Art. 56.
dx
84. To integrate _. In the formula of integration
f udv = uv fvdu, put v =x, =-..du =-- (x~ a) Then
+) x Ia/ dx
__ + + a(X 2S 2
(X 4 a) (x,. a2) -t- 2 p akl 4- a2)P+
Whence, transposing, putting p + 1 =n,
+ 1 x 2n - 1 dx
E2xn- 2 (x" a n)1L-' -n - af J(x a-)L-'
Except when n = 1.
FORMULAE OF REDUCTION.
47
When n is a positive integer, this formula of reduction reduces the integral ultimately to --- =- tan-1 - (when
d +ad a
a2 has the positive sign). When a2 has the negative sign,
the ultimate integral is - = -logx -
SJ -- a 2a x -- a
/ (A + Bj) dx B p (x + Qb)dx
85. To integrate -,
+ (A - Bb) (+ + 4 b4 c)'=
(Art. 44, except when n = 1),
-B
2 (n-1) (X+9 +bX + C)
dx
S {( bY)2 + (c - )"
-B
2(n - 1) (T" + 'bx+ c)"-l1
A - B b x_+b
2n-, (C - 2) {(x + b) + (c-b2)}"-1
3 1 A - B b)d, n -- N c - b {( + b) + c - b}"by the last article, putting x + b for x, and c - b for a2. All
the constants may be positive or negative.
When n = 1, we have from the first equation of this
article and Arts. 43 and 59,
r A + Bx)dx B
I bx + c = log, (xv +2 b + c)
A - Bb x + b
+ (c- btan- (" ( -
INTEGRAL CALCULUS.
SECTION VIII.
RATIONAL FRACTIONS.
86. A rational integral function of x is the sum of a finite
number of terms which involve only positive integral powers
of x, and these as factors.
87. A. fraction rational with respect to x is a fraction of
which the numerator and denominator are rational integral
functions of x.
88. The partial fractions of a given rational fraction are
those rational fractions with different denominators of which
the sum is equal to the given fraction.
89. If the numerator of a rational fraction, cleared of
negative indices of x, be of higher dimensions in x than the
denominator (i.e. contain higher powers of x than the denominator), the fraction may be reduced to a rational integral
function, + a rational integral fraction of lower dimensions
in the numerator than in the denominator.
For if a rational function of x, axp4-q + bp+q-1 +... be
actually divided by another such function of lower dimensions
in x, AxP3 + BxP-1 + CxP-2 +... (p and q being positive
integers), it will be found that the quotient consists of terms
with descending positive integral powers of x, commencing
with the index q, and ending with the index 0; and the
remainder, after division, has terms with only positive integral powers of x, commencing with the index p - 1, and
ending with the index 0. So that
axp+q + bXp+q-I + CXP+9-2...
AxP + BxP-1 + CXP-2 +...
axP-1 + bxp-2 +...
An 4" + B I-1-... - -------
A xP + BxP-1+...
where the coefficients A, B,... a, b... are to be determined
in the course of the process of division.
RATIONAL FRACTIONS.
49
90. The rational function A, xQ + B, x-1 +... is immediately integrable by Art. 44. So that for the complete
integration of a rational fraction, all that is required is to
integrate a rational fraction of which the numerator is of
lower dimensions than the denominator.
91. If in any rational integral function of x, x* be assumed to have the value b x + c, the function becomes linear
(i.e. of one dimension in x). For x = x'. x = (bx + c)x
by the hypothesis; = bx2 + cx, which again, by the hy
pothesis, is equal to b (bx + c) + cx, which is linear.
So, likewise, may xa, x, &c., be reduced to a linear form.
So that any rational function of x takes the linear form.
ax + r?,
when bx + c is substituted continually for x; a and 9 being
quantities not affected by.92. If the preceding a+ = 0 (1), then a = 0 and p = 0.
For the original assumption x = b x + c, gives x=
S{b + (62 + 4c)~}, and x =- {b - (b- + 4c) }. Therefore
equation (1) is required to be true for two different values
of x (except when 4c = - b2); call them z, x, Then
acx + 3-= 0
a + - = 0.
Subtracting, a (x, - x) = 0,.. a = 0, since x, - -x is
not zero.
Substituting a= 0 in either of the equations last written,
we get 0 = 0.
93. To show that real quantities, A and B, independent of x,
may be found such that
<x _ Ax 4- B
= x + xx...... (
(x - bx - c)" +x (X26 - b - c)"
where O x and 4x are rational integral functions, and do
not contain x2- bx- c as a factor, xv a rational fraction,
and n a positive integer.
S- (Ax + B) /,x,
(x A, XX.......................... (.)
(X2 - bx - c)n *X
Now, by a principle proved in the theory of equations, any
D
OU INTEGRAL CALCULUS.
rational integral function of a contains a1 - bx - c as a
factor if the function = 0 when x2 - bx - c = 0.
The numerator on the first side of (2) is a rational integral
function of x. If, therefore, real quantities A and B can
be determined, so that this numerator = 0 when x- bt -
c = 0; then the numerator is divisible once, at least, by
a'- bx -c.
The quotient will be a real rational integral function,i t.
Then (f) becomes
( b x9) x.................. (3.)
(X - bx - c)"-' +X
or xx is a rational fraction.
It only remains to be shown that A and B are real quantities, when determined by the condition supposed, namely,
that
OX - (Ax + B) +x = 0... (4), when x" - bx - c = 0.
It has been shown by the last article but one, that when
x, - b - c = 0, or x2 =- bx + c, cx is reduced to the linear
form ax + /, and +x to a similar linear form a'" + - ',
where a, 3, e', A', are real quantities; therefore, (4) takes
the form
ax + 3 - (A x + B) (a'x + ') = 0,
or, multiplying the quantities in parentheses, and putting
x- = bx + c,
ax + - - A a' (bx + c) + a 'x t+ (a'ax - + ) = 0.
By the last article the coefficient of x in this equation is
zero, and the quantity independent of x is zero, or
- A (a'b -- i) + BC' = C,
3 - Aa'c + B 0' = 0.
(Except, as before, when - 4c = b", when (x" - bx - c)"
= (x - b)2'"; see next article but one.)
It is clear that the values of A and B found from these
equations are real quantities, independent of x.
From (1) and (3),
ax Ax + B
(x2 - bx - c)" ' - (xt - bx - c)"
+( - b - 6 )"-,................ (a)
RATIONAL FRACTIONS.
51
94. Supposing the last fraction in this equation in its
lowest terms in a:2 - bx - c, we have, similarly,
(), X A,__ + B
(X'2- -x-c)"-' *x (, -bx-c) " '
+ ( x
(x'' - bx - c)" - '
and so on. Therefore, generally,
() Ax + B
(x - bx - c)'L x (x - bx - c)"
A,x +. B, A,x + B,, ex
+.o... +
(+ ' - b - c)"- x - bx - c b'
where 4 x is a rational integral function of x.
95. To shew that a real quantity, C, independent of x,
may be found such that
_ x C
= + XX.................. (1.)
(x - a)"nx (X - a)"
where (px and /x are rational integral functions of x, xx a
rational fraction, n a positive integer, pqa not zero, and +,a
not zero.
)x - c+x
(x a)" +/ta
Let C ==,-(which is finite by hypothesis).
/a
Then u(x - 0-, the numerator of the fraction on the
*/a
first side of (2), is zero when x - a is zero; and, therefore,
is divisible by x - a, once at least.
Then (2) becomes
(x - at)o"-n ' Ix,
From this equation and (1),
(ax C 0,x
CtLn * X,......... (_X)
(X - a)" + ( -- a)" (x - ) )'L-' ~x X
D 2
INTEGRAL CALCULUS.
96. If the last fraction in (i3) be in its lowest terms with
respect to x - a, the numerator does not contain x - a, and
(p a is not zero. We, therefore, proceed as before, and put,x__ C, C x
(X - a)-I - (aX _ n- +- a)12 (+ -X
and so on. Therefore, ultimately,
X+...... + C,
(x - a)"# = (a - a)' (a; - a)71 +1.x
97. In the formuhe marked (a) and (i) in the last article
and the preceding, respectively, the numerators px,,x, px,
&c., have been supposed not to contain the simple or quadratic
factor expressed in the denominators. If, however, either of
these numerators happen to contain any number of times
a factor of its denominator, reduce the rational fraction by
division by the factor that number of times, and proceed to
reduce the resulting fraction into its partial fractions.
98. If the quantities U,, U2... represent quadratic, and V,,
V2... simple factors, we have, by the last two articles, continually reducing the rational fractions into partial fractions,
Ox
U,I U2...* V, M V2M
Ax+B Ax + B, A xn, + B,
--.- + +... +
U, I U196 - + U1
A' x + B' A'x, + B,'/ An,2x + B'n,
+ + -... U,
+ &c.
C C, Cm,
+ +--+... +4 -V 2i V1l - 1 Vi
C' C' C'1t
C/ / I +.. M +
V2 +O 1 V2
+ &c.
RATIONAL FRACTIONS.
53
99. In resolving rational fractions into partial fractions,
the greatest difficulty occurs in those cases in which there are
quadratic denominators of the partial fractions, and their
numerators are therefore linear in x Where, however, the
partial fractions have only simple denominators, there are
no (A)s and (B)s, and the numerators (C) are easily found
by either of the following methods.
(1.) Clear the equation of the last article of fractions, by
multiplying by the denominator of the first side. As the
denominator is supposed to contain no quadratic factors, it is
equal to V/"1. V2"2..., and therefore is of m, + m, +...
dimensions in x. Therefore, when the equation is cleared
of fractions by multiplication by this denominator, there are
terms in the second side of the resulting equation of (m, + m2
+...)- 1 dimensions in x. The new equation contains,
therefore, (m, + mz, +...) different powers of x, and (equating
coefficients of those powers) there are therefore mi +, +...
equations to find the m, + mz +... quantities (c).
1 1
EXAMPLE.-TO resolve = - -
x - - x + I (x - 1)" (x +1)
into partial fractions. Assume
1 C Ci C2
(x - )(x + 1) - 1 (x - 1) x + 1
Clearing the equation of fractions
1 = c (2 1) + C, (x + 1) + C (X- 2 + 1)... (a.)
Equating coefficients of x2, 0 = C + Ct
S of x, 0 C, - ýC2,,,, of x, 1= - C + C, + C,.
Adding these equations, we have 1 = QC,,.. C, -.
Substituting this 'in the second of these equations, we have
C2 = ~, and therefore, from the first equation, C = - 4.
f dx 1 dx 1 f dx
J - x 4 x-1 - 1)
1 dx 1 1 1 1
S - 1 log (x - ) - 2 +- log (x +1).
4 x + 42 x - 1 4
INTEGRAL CALCULUS.
(2.) The numerators of the simple partial fractions may
be found by another method, which is frequently more convenient than that of equating coefficients. In the equation
cleared of fractions, give x successively the values which make
each of the (V)s zero. Then, in each case, all the (C)s
disappear but one, which is therefore determined.
For instance, in the equation (a), in the last example, put
x = 1. Then (a) becomes 1 = C,. 2 or = C,.
Put x = - 1. Then (a) becomes
1 == C. 4, or C, = -.
100. By this method of substitution, it is clear that as
many coefficients (C) are determined as different simple
factors of the denominator of the fraction to be resolved into
partial fractions are made zero. But when this denominator
contains higher powers than the first of any of its factors,
there are more (C)s to be determined than there are different
factors. For instance, in the example just considered only
two different factors x - 1 and x + 1 can be made zero, and
therefore only two out of the three (C)s can be thus found.
To determine the remaining (C)s, differentiate each side of
the equation equivalent to (a) in the last example; for since
that equation holds for all values of x, the differential coefficients of the two sides of the equation are equal.
In the new equation obtained by differentiation, put the
factors = 0 successively, and so obtain more values of (C)s.
Then, if necessary, differentiate again, and equate factors to
zero, and so on continually, till all the (C)s are found.
For instance, in the last example, differentiate (a), then
0 = C. 2x + C, + C2.(a- 1).
Put = 1. Then
0 = C. + C,,.. since C =-, C= ---.
101. We will take, as another instance, a fraction to be
resolved of which the denominator contains the third power
of a factor, and which therefore requires two successive
differentiations.
2 X- + 1 C C, C, c
(x - ) (X + 3) x - ( l x- 2) + (x- X +,3
_cl
+ c+
(x + '3)
RATIONAL FRACTIONS. 55
Clearing this equation of fractions,
X 9 + 1 = C (X - 2)2 (X + 3)2 + C, (1 - 2) (X + 3)2
+ c ( + 3) + (X - ) (x + 3) + c1 ( - )... (a.)
Putting x =, 9= C2. 25,.. C =19
x=- 3, 19=c,(-5)",.'. o, == 5:
Now differentiate (a).
4x = C {2 (x - 2) (x + 3)2 + 2 (x - _)2 (x + 3)}
+ C, (x + 3)2 + 2 (x - 2) (x + 3)} + C, 2 (x + 3)
+ c {3 ( - V) (x + 3) + (x - })3} + c 3(x - 2)2... (b.)
S. 9
Putting x = 9, 8 = C '5 +- C. 10,.". C,= 5, since C = - 2
57
x = - 3, - 12 == c ( - 5)* + c 3 (5), = c(- 5)3 -19 1 (57 1 3
since c,= -,.'*. 5 5 -1
Differentiate (b), retaining only terms which do not vanish
when x = 2; then
4 == C 2. (x + 3)' + C1 {2 ( + 3) + 2f(x + 3} + C.. 2,
x being supposed = 2. Consequently,
3
4= C. '2. +C,.(56+5)+5C,.*.C=-5
2 x + 1 3 ~22 9
2 + +
(w - 5) (x + 3)2 54(x - 2) 5x(X - )2 25 (x - 2)
3 19
+ 54 ( + 3) 5 (x + 3)'
as may be verified.
102. Where the denominator of the fraction to be resolved
contains quadratic factors (and especially where each such
INTEGRAL CALCULUS.
factor is trinomial (= x2- b - c), the difficulty of resolving
the proposed fraction is considerably increased. The student
will probably be inclined to think that considerable labour is
saved by the following method, if he will compare the amount
of work which it requires for a difficult example with the
amount required for the same example by other methods
which have been proposed.
Assume the proposed fraction to equal a series of partial
fractions, as in Art. 96. Clear this equation of fractions, and
so obtain an equation corresponding to (a) in the last examples.
In this equation make each quadratic factor x2 - bx - c = 0
(i.e., substitute b x + c for x). Then the equation may
be reduced to the linear form ax + - = 0 (Art. 91), and
a = 0, 3 == 0 (Art. 92). From these two equations the A
and B corresponding to the factor xa - bx - c may be found.
This method will give as many different (A)s and (B)s
as there are different quadratic factors, successively made
zero.
If there be more (A)s and (B)s (i.e., if any quadratic factor
appear in (a) of higher power than the first), differentiate(a),
and in this derived equation make all the quadratic factors
zero successively, then, if necessary, differentiate again, and
in the second derived equation make the factors again zero,
and so on continually, till all the (A)s and (B)s are found.
The (C)s, if any, corresponding to simple factors, may be determined from (a), and the derived equations by the method
already explained.
Let us take, first, an instance of the simplest case, that of
quadratic factor, which wants its second term, and is therefore
binomial.
x3 dx
103. To integrate X x
(_ - 1)2 (X2 + 1)
XAssume_ Ax - B C C,
Assume -t +
(xz 1)2 (X2 + 1) x2 + 1 - ( -1)2'.X. = (Ax + B) (x - 1)2 + C (x2 + 1) ( - 1)
- C, (t + )... (a.)
First, to determine the (C)s by the method of Art. 98, let
x= 1,.". 1=. 2, orC =.
RATIONAL FRACTIONS.
Differentiating (a), and for brevity retaining only terms
which do not vanish when x = 1, we have then
3x2 = C(x2 + 1) + C12x,
where x = 1. Consequently 3 = C. + C1. + 2, or
3
C --= - C- 1.
Secondly, to find A and B by the method of the last article.
Make the quadratic factor zero in (a); i. e. put - 1 for x2
continually; (a) becomes (expanding (x - ])2 and putting
x" x. x2 = X)
- = (Ax + B) (- 1 - 2x + 1)
= 2 A - 2 Bx (putting - 2, Ax,2 = A),.. = 2A -(2B- 1) x,
which is of the linear form required by Art. 91. By Art. 92
the coefficient of x in this equation, and the quantity independent of x are each zero;.* A = 0; 2 B- I = 0, or
B = 1. Hence,
S (x - 1) (X2 + 1)
1 1 1 1 1
- 52 + 1 x-1 2 (x-1)2
x xdx ]1 1 1
1) (+ ) = an log(- )- x -
Next take a case in which all the operations for resolving
partial fractions are required, and the quadratic factor is trinomial, and raised to a higher power than the first.
x.+ 3x - 2
104. To integrate x2 x - - 0---1. Assume the
(X - X + 1)2 (x - 1)
fraction -- A ~ + A x + B1 C C1
x2 - x+ 1 (x - x +-1)2 x - 1 (x- 1)
x2 + 3x - 2 = (Ax + B) (x2 - x + 1) (x - 1)2
+ (A, x + B,1) (x - 1)2 + C (x2 - x + 1)2 (x - 1)
D3
INTEGRAL CALCULUS.
First, to determine the (C)s, Art. 100. Put x= l,... =CIO
Differentiate, retaining only terms which do not vanish
when x = 1,
x~3=C(d2-2X+1)~C15.(Q -1) X+1),
where x=1,.. 5= C + 2C,.. C=]l.
Secondly, to find the (A)s and (B)s, Art. 10%, put x' = x- 1
continually in (a); (a) becomes (x - 1) + 3 x - 2 =(AI x + B1)
(x-1- 2 x + 1) = (A + B1) (-x)= - A (x - 1) -B x,,
or 0= 3+A -x(A, +B,+4), whence Art. 92, 3+A =0,
or A =- 3. Also A - B, + 4 = 0,.. B= - 1.
Now differentiate (a), retaining (for brevity) only terms
which do not vanish when x - x + 1 = 0,
2x + 3 = (Ax + B) (5x - 1) (x - 1)' + A, (x - 1)2
(A~x + B,) 2 (x - 1),
when x = x - 1. Making this substitution continually,
to bring the equation to a linear form, we have, since
(x - 1)2 = - a,
2x +3)= {2A(x - 1) - Ax+ B (2x-1)} (-x)
- 2 A, x + 2A (x - 1) + BQ, (- 1)
= (Ax 2Bx- 2A-B)(-x) - Ax 2B, x-2A, -2BI
0 =- 5a - 3 - (A + 5 B) (x - 1) + (2A + B)x
- (AI - 2B1) x - 2A1- 2B1.
This equation being of the required linear form, make
the coefficient of x and the quantity independent of a each
= 0. Art. 99.
0=-0,+A-B- A-+BBA,., A-B=1,
0 = - 3+ A+f+B - 5 2Aj-QB,.. A+ 2B= - 5,
B = - 2, A = - 1.
Hence the proposed fraction is equal to
x_+2_ 3a~1 1 ~
z- a+1 (a- +1) a-1 + (x-f) '
RATIONAL FRACTIONS.
59
f(x~ 2)x 5 x 2Ix - 1 +
- ta n- + - log(x - x + 1)
13 2 '
Art. 85.
(3 + 1) d - 2x -1
(x - x + 1)" 2 (d - xm + 1)
5 2 Ox 1
+ -. _tan-' -, Art. 85.
3 /3 2/3
_ log ( - 1)
Jx) -1 Art. 44,
( 1 - X ) dm 1
3+ ox - 0
x2+3x-2 7- 5x +.~~d *tl dx.4..52 ~
(Xd x + 1)" (m - 1)2 3 (' - X + 1).
25 S -- 1 m -
tan- I + log
V3 3 (X- x+ 1)&
INTEGRAL CALCULUS.
SECTION IX.
RATIONALIZATION.
105. THE last method of reducing functions of one variable
to integrable forms which we have here to consider, is the
method of Rationalization, which is a system of algebraical
substitution, by which, for an irrational algebraical function,
is found an equivalent which is rational, and therefore
integrable by the preceding section.
106. A rational function has a rational differential coeficient. Every rational function of z may be reduced to the form
* a + bz + c2 +... kz
a + bz + cz +... Iz '
and it is clear the differentiation of this quantity cannot
introduce fractional indices of z. It follows, that if x be
dx
any rational function of z, is a rational function of
z =R; suppose,.'. dx = R.. dz, where Ra is a rational
function of z.
107. A rational function of a rational function of x is a
rational function of x. For if c, Jf both indicate rational
functions, fx involves only integral powers of x, and ) (fx)
involves only integral powers of fx;.'. 0 (fx) involves only
integral powers of x, or is a rational function.
108. A universal method of rationalization cannot be given,
as many irrational expressions are reduced to rational forms,
by artifices peculiar to the cases in which they are applied.
But the most general principle of rationalization may be
stated as follows:Suppose that the expression to be rationalized is a rational
function of an irrational function (I,) of x, and of a rational
function (R,), so that the expression to be rationalized is
f (I, RX);
RATIONALIZATION.
61
where f indicates a rational function. Then assume, if
possible, x equal to such a rational function of z, that 1X
becomes equal to a rational function (R.) of z. Then also,
by Art. 106, dx = R' dz. Also, by Art. 107, Rx = R",
another rational fnnction of z;.'. f (I,, Rx) dx =/(Rz, Ra',) Ritdz.
But f indicates a rational function. Hence, by the article
last referred to, f(R,, R") R'zdz is rational in z, or
f (Ix, Rx)dx is reduced to a quantity which is rational,
and therefore integrable by the methods of the preceding
section.
m
109. To rationalize R, ax+- )i dx, where RX is a
\ ]a,.- + bi
rational function of x and mn, n positive or negative integers.
This is a particular case of the last article.
Letax + b a - a, ("
Let -----= ^, ". x=-_--,......... (1),
aix + bi b - blzn
or x is a rational function of z. Then by the last article,
( ax+ 4 Wb\n
RX =R"z, dx=Rdz, 1,- =a+,
aix +
and so the whole of the proposed expression is rationalized.
110. To rationalize (a'x + b')1' (ax + b)'dx, where one of
the three quantities
I,, v, or /i + v is a positive or negative integer... (2.)
In the expression proposed to be rationalized in the
last article, put R- = (a'x + 6)', where i is a positive or
negative integer.
Put a, = a', b = b'. Then the expression becomes
(a'x + 6')" (ax +- b)" d(x,
which may be written
(a'x + 6') (ax + b)dx,
INTEGRAL CALCULUS.
where p + v (= i) is an integer, or (2) is satisfied; and by (1),
ax +b6
=.......... (3)
a'x + b'
Next, let Rx = (a'x + b'), and in I, let a, = 0, b = 1.
Then the expression rationalized becomes
(a'x + b') (ax + b) dX,
which, again, is of the form
(a'x + b')E (ax + b)V dx,
where one of the quantities t or v is an integer, and the
condition (2) is satisfied. In this case (1) in the last article
becomes
zn _ b
ax + b = X, -....(4)
111. To rationalize xr (axq + b) dx.
1 1 L-1 1'
Put x = x, '. -.x dx = d, xP xq, and the
q
expression proposed to be rationalized becomes
p1
1 +1-1 m
- x q (ax + b) dx.
q
This can be rationalized by the last article, whenever
S+ -- 1 is an integer, and, therefore, - an integer;
q q q
1 m _.p+ m
or + -- 1 + - an integer, and, therefore, -P -
q q, n q n
an integer.
The First Criterion of rationalization of
xP (axq + b)t dx,
is, that p+ be a positive or negative integer, when
q
(since x7 = x) we have to assume a.T + b -- z" by (4).
RATIONALIZATION.
63
p+1 m
The Second Criterion of rationalization is, that -- + -
q n
be a positive or negative integer, when we have to assume
ax< + b
S = z~ by (3).
x7
112. The method of Art. 108 may be extended to several
irrational functions 10', I, 1I)... if it be possible to assume
x such a rational function of z, that these irrational functions
of x become equivalent to rational functions of z.
For instance, if the irrational function of x be
1 1 1
(a+ bX a + bx a+b &x d
fa~a/ \a+ &C.dx
( a + bx 'm+i a, + bjx a' + bx, d.
where m, n, &c., are integers.
a + b - Zp' - a-.'."*
Put - m-+I.... V na, + bi n b -- b R...
I 1(12)* I = Z n e*** _() = z"nQo*
dx is rational in z; and so the whole expression may be
rationalized.
INTEGRAL CALCULUS.
SECTION X.
INTEGRATION OF FUNCTIONS OF SEVERAL VARIABLES.
113. WE have hitherto considered the integration of functions of only one independent variable. The magnitude of a
quantity may, however, depend upon the magnitudes of several
other quantities, each of which is susceptible of independent
and separate variation.
For instance, the cubic content of a right cylinder depends on two independent magnitudes, the altitude and the
area of the base. Each of these magnitudes may be considered to vary independently of the other, for we may
conceive the existence of any number whatever of cylinders
with equal bases but different altitudes, and of any number
of cylinders of equal altitudes but different bases.
Again, the content of a rectangular parallelopiped is a function of three independent variables the lengths of three of its
edges. The content of an oblique parallelopiped is a function
of five independent variables, namely, the lengths of three of
its edges, and the inclinations of two of them to the third.
The weight of a solid is a function of two independent
variables, its volume and specific gravity. The time of
vibration of a perfect pendulum vibrating in vacuo is a
function of three independent variables-its length, the force
of gravity, and the extent of the oscillation.
114. DEFINITION. The Quadrature of a finite continuous
function of several independent variables having a limited
range of values, is the sum of a series of different values of
the function, each multiplied by the differences between the
corresponding values of all the variables and their next preceding or succeeding values.
115. The Multiple Integral of such a function is the limit
which its quadrature has when the differences of the independent variable approach zero, and their number infinity.
[These definitions are extensions of those of Articles 16
and 17.]
INTEGRATION OF FUNCTIONS.
116. Letf(z, y, x, w...) be a finite continuous function of
any number (N) of independent variables. Suppose nI values
given to z, n, values to y, n, values to x, &c. Then the
total number of different values of the function will be the
total number of different combinations of n1 + n, + n3 +...
different things taken N together.
Let Z, z, Y, y, X, x... be the superior and inferior limits
of the several variables. If Y be understood to be the
abbreviation of the words " sum of terms of the form of,"
the quadrature of
f(Z, xj, w, s....)= =-f(, x, w...) Z.. Y. x.8w...
where 8z, by, x, 8w... indicate differences between successive values of the variables. Also,
limit of if(z, y, x, w...) z. y. x. 8w...
(when bz, by, bx, zw... approach the limit zero), is equal to
the multiple integral of f(z, y, x, w...) between the limits
Z, z, Y, y, X, x... This multiple integral is written
Z Y X... f (Z y X w.) dzdydxdw...
the sign f being repeated as many times as there are independent variables.
117. Multiple integrals found by successive integrations.
Let zZ, 2, z2... y, iy, '..'3 &c., be successive intermediate values of the variables between their limits. Also,
let 8z,, Bz2, 83... by1, by,,... &c., denote the successive
differences of the values of the variables. The integral is
the limit of the sum of terms of the form
f (Z., X., -,...),. I na ' -X,...
First. The sum of the terms in which z alone has different values, while the other variables have their first
values, is
{f(z, y7, x1...) bI + (,2, Y, I...),2
+ f(zA, 1, X1...) - z +...} i 81...
of which the limit (since here z alone varies) is equal to
limit of by1 8,... f (z, y, x,...)dz.
00 INTEGRAL CALCULUS.
This integral being taken between limits, involves only
those limits, which may be functions of x, y,... or any other
quantities whatever. But the variable intermediate values
of x disappear (Art. 26) from the integral, which, therefore,
takes the form f, (yl, xl, w1...), z being omitted.
Secondly. Add all the terms in which z alone varies, y
having its second value, x, w... as before their first values.
The limits of the sum of these is
limit of y. ax... / (z, yf, xi wl...) dz
= limit of y,2.. Y. fA (2', X, w..*).
Similarly for the terms is y &, &c. The sum of all
these is
{i(Jf, &,X1,.. ) BY + /(.,, l...) 8'Y
+ f(y,,,...) a3 +...} xi. r...
of which the limit is (by reasoning with respect to y similar
to the preceding with respect to z) the
limit of 8Ex. awl... f ( 1...)d
= limit of x1 wl... f (x1, c1,..),
y being omitted from f2.
Continuing the process, x, w... successively disappear
by successive definite integrations; and the final result, or
required multiple integral, is the result of as many successive integrations as there are independent variables.
Hence, where there are only two independent variables,
if r be the last of the independent variables, this result is
of the form
Sfr dr = F (R)- F ().
f f(z, y) dz dy = dy{Jf f(z, y)dz;
fz zf =f
INTEGRATION OF FUNCTIONS.
where there are three independent variables,
f Y/ f (,, x) dz dy dx
fx d [f y d4 (zf}, ]) dz
And, generally, a multiple integral is formed by integrating the proposed function with respect to one variable,
as if the others were constant; substituting the limits of that
variable; integrating the result with respect to another variable, as if the rest were constant; substituting the limits,
and so on, till as many integrations have been performed as
there are independent variables.
118. Order of integration indifferent. The sum of any
number of quantities does not depend on the order in which
they are added. Hence in the summation of the quadrature,
the terms involving different values of any variable may
be first collected, and the limit of their sum involves an
integral with respect to that variable. Therefore, the variable with respect to which the first integration is performed,
is indifferent. Similar reasoning applies to the other integrations.
COROLLARY. f dy ( f(zy) d)
=-f' z ff, y) dy)
119. The cubature of solids affords a very complete illustration of the foregoing principles.
Let xOz, xOy, yoz be three planes perpendicular to
each other; and let ABCDabcd, be a solid bounded by
the curved surface ABCD, by a rectangle ac in the plane
xOz, by two planes Ab, Dc parallel to the plane yOz,
and two planes Ad, Bc parallel to the plane xOz.
Consider now the base ac of the solid divided into any
number of rectangles, represented by dotted lines in the
figure, and on these rectangles, as bases, let rectangular
parallelopipeds be described, of which the sides cut the upper
surface A BCD in the curves shewn in the diagram.
Ob INTEGRAL CALCULUS.
If x, y, be co-ordinates of any point (P) in the curved
surface referred to rectangular axes o x, Oy, 0 z, the relation
between a, y, z may be expressed by an equation
S= f (, y),
in which z is supposed to be finite and continuous;
and pq = x, Oq = y, Pp = z.
z
Let Pp be the altitude of one of the elementary parallelopipeds, x and oy the length and breadth respectively of its
base. Then the solid content of the parallelopiped is the
product of these quantities, or zhxvy =f(x, y) Sx. ýy.
Let o x, x......
y^---zir-- i--z^^
Yo, Y19 V2.. yn,
be corresponding successive values of the co-ordinates, and
sx, xy, the common differences of the successive values of
x and y respectively. Then it may be seen that the solid
Ac contains parallelopipeds, of which (reckoning them in
rows parallel to ab) the solid contents are
f (~x, Yl) xy, f (X -/7)2xy, f (x, y/) ~ fy...f(x, Y,) ),
f(X, yi)44 fX,/( Y,) ) /(4, f(I)', yY) ýx...f Y(x",,) Sy.
INTEGRATION OF FUNCTIONS.
69
Also, as will be proved hereafter, the more the number
of these parallelopipeds is increased, and their length and
breadth diminished, the more nearly is their sum equal to
the content of the solid AC. If the limits of the sums
of the contents just written be taken in rows across the
page, the result is
limit { x f Yof (x, ) dy + x /Of(x, y)dy+...
+ 8 Yo f (,,,x y) dy}
S 0xo { Vf (x, y) dy t d y.
If, however, the parallelopipeds had been reckoned in rows
parallel to the longest side of the page, that is, parallel to
ab in the diagram, the limit of the summation would be
f yo { of xf, y).yxY dy.
And since both results represent the same solid content,
they are equal.
INTEGRAL CALCULUS.
SECTION XT.
QUADRATURE OF CURVES.
120. THE Integral Calculus is applied to the rectification, or
determination of the lengths of curves; to the quadrature, or
determination of areas of curves; the complanation of surfaces, or determination of their superficies; and the cubature
of solids, or determination of their volumes or contents.
121. The methods of determining Quadratures and Cubatures are readily demonstrated by principles already laid
down. Rectification and Complanation depend on geometrical
theorems, hereafter given.
It has been shown, Art. 19, that if a and y be the rectangular co-ordinates of any point of a plane curve, x, Y,
and x, y the co-ordinates of its extremities, the area included
by it, and straight lines from its extremities parallel to the
axes of x and y respectively, is given by the formulae
/ xdy, or J y2 dx,
j 6X *J y
where it is supposed that
x and y are always positive Y
and finite, and to neither
is assigned more than one
value corresponding to any
value of the other, between
the limits X, Y, x, y.
122. Quadrature of the
Circle. Let r be the radius
of the circle; x, y, its coordinates at any point referred to the centre as origin of co-ordinates; then x
and y are connected by the o
equation.
b-----------------
b e
QUADRATURE OF CURVES.
71
x2 + 2 = r2;
or, y = (r - 2),
Jydx =, (- x)dx
= - 2) + (dx (integrating by
r - )" parts),
Now (r - x) dx = (r2 - x) x
+ r:I di r) - f;; i
+ 2 /--- / --- z jX.
The last integral on the second side of this equation is
identical with the integral on the first side. Therefore,
transposing and integrating the remaining integral by Art. 56,
S- x) dx = x (r2 - x2) + r2 sin-1.
If Oc = X, and ob=x, we have to take this result
between limits X and x, to find the area Abc;.. Abe = X (r - X2)- x (r2 - x2)
X x
+ - r2 sin1 - - X r2 sin-1
r " r
If it were required to find the area of a quadrant, B,
C would be supposed to meet Oy, O x, respectively, and therefore X = r, x = 0. Therefore, since sin-' 0 (or the angle
7T
of which the sine is 0) = 0, and sin- 1 = -,
quadrant = - r.
Therefore, area of whole circle = r r2.
123. Area of Ellipse. The equation to the ellipse referred
to the major axis, and a line at right angles to it at its
extremity as axes of co-ordinates, is
INTEGRAL CALCULUS.
y = - (20ax - x2)I,
where a is the semi-axis major, and b the semi-axis minor.
0 B -- - - - x--- --.
ydx = (2a x - (2ax dx
a a\
a-- ab Cos-' (Q ax
When x = 0 the preceding expression vanishes. It may,
therefore, be supposed to be taken between the limits 0
and x; consequently, if OB = x, the expression is the value
of the area PBO.
When a= b the ellipse becomes a circle, and the expression (1) for the area becomes
a~ - - X ---
a cos--- (Q a x -,6)1...... ()
Hence, if OP'M be a circle having the same centre C with
the ellipse OPM, and OM, the diameter of the circle, be also
the major axis of the ellipse, we have, comparing (1) and (2),
area OP'B a
area OPB b'
It appears also from (1), that the area OPB is proportional
to b. Hence, if any number of concentric ellipses were
QUADRATURE OF CURVES.
devcribcd o t he SRam
tbh saim. bas,
several JmilPr t s
axis 1111i - e 01 ea' of them having
ouli le li the1 ip)oportion of the
The area of a quadrant of the ellipse is found from (1),
by putting x = a, to be
3 ab cos-, 0 = a.
4
Hence the area of the ellipse = 7r a 6.
124. Quadrature of curves referred to oblique co-ordinates.
The method of obtaining, in Art. 19, thle quadrature of curves
referred to rectangular co-ordinates, consists in dividing the
74
INTEGRAL CALCULUS.
area by rectangles, and taking the limit which their sum
has when their breadth is indefinitely diminished and their
number iridefinitely increased. =
Similarly, if an area, ABCD, bounded by the curve BC, and
three straight lines, of which BA is parallel to CD, be divided
by parallelograms upon AD having sides parallel to CD, the
limit of their sum is the area AECD. Also, let the curve be
referred to oblique axes of co-ordinates Oy, Ox, inclined to
each other at an apgle a. If lx and y be the lengths of two
sides of one of <he parallelograms, y sin a is its altitude,
and y sin aýx is its area; whence it is easily seen, that the
area ABCD =Jy sin adx, taken between proper limits.
125. Quadrature of the Hyperbola. Let the hyperbola, of
which A is the vertex, be referred to its asymptotes Ox, Oy,
0 B M
inclined to each other at an angle a, as axes. Draw AB
parallel to Oy, and let OB = e. The equation to the hyperbola is y x = e. Om = x.
x 1x e2
Area ABPM = sin a J ydx = sin a x dx
x
= sin ae' log-.
e
126. Quadrature of the Witch of Agnesi. In the last
example, as x increases, the area increases indefinitely; and,
therefore, the whole area between the curve and the asymptote
is infinite. There are, however, curves in which the area
between an infinite branch of the curve and its asymptote are
QUADRATUJRE 0! CURVES.
finite. The "witch," or "verl a" of
Donna Maria Agnesi, is an ti, e. c
Let AB be a diameter of a cirlet =;,
AC a tangent, P any point in the i.:re, r
AM =-x; AB, AC being the axe lJ,
and y respectively.
The curve is defined by the relahi,
rectangle PA = rectangle DB.
The equation to the curve will N
found to be xy' = a2 (a - x).
a -x) a-x A
X (ax - x
a - aB A
a- x
= a(ax - x') + ' a2 c I a
Arts. 44 and 56.
This expression is to be taken between limits x =a and
x = x, to give the area PBM.
The area between AC, AH, and the curve, is the limit
which the result thus obtained has when x has the limit 0.
This evidently is found by taking the expression for the
integral between limits x = a and xv = 0;.. required area = {cos- (- 1) - cos-11} a = 2 tra.
The whole area between the asymptote and the whole
curve on both sides of AB, is double the preceding, or = 7ra2;
and, consequently, is four times the area of the circle.
1Q7. Quadrature of the Cissoid of Diodes. This curve, invented by Diocles, a Greek mathematician, about the sixth
century, and used for finding two mean proportionals, resembles the curve last considered in several respects. It
affords another instance of a finite area included between an
infinite curve and its asymptote.
The cissoid may be defined by Newton's method of tracing
E ý
76 INTEGRAL CALCULUS.
it. The arms of a bent lever are at right angles to each
other, and the end of one of them slides along a straight
line, while the other is always in contact with a point of
which the distance from the straight line is equal to the
length of the first arm. The angle of the lever traces out
the cissoid.
A
Let B be the fixed point. Then, if AP = BD, and the
end A of the lever move along a straight line, while PC
remains in contact with B, the cissoid is the locus of P.
Let AC = a, AB = x, PB =y. The H
equation to the cissoid will be found to be
y2 (a - x) = x3.
3 dx
fydx =J= ( _)___
AC
-= (a - _x) v. + 3f(a - x) xdx
(integrating by parts).
Also, (a - x) x dx = (ax - x2) dx
= {(4 a) - (x - -a)}idx,
which is of a form which has been already
integrated (Art. 83);
QUADRATURE OF CURVES.
77
+ 3 ( x - a) (ax - x2)l + ~ a2vers-1i--.
"2
For the whole area between AC, CH, and the curve, it
appears by the same considerations as in the last article, that
this integral is to be taken between the limits x = a and
x = 0, when
fydx = 2 a2 { 2 - v- ers1 0} = a2r.
The whole area included by both branches of the curve
and the asymptote is double this, or _ 7ra2 = three times the
area of the circle of which AC is the diameter.
128. Polar co-ordinates. Let the position of any point in
a plane curve be referred to polar co-ordinates, namely, the
length (r) of the straight line
drawn from the point in the Y
curve to the pole, an assigned
point in the plane of the curve; r
and the inclination (0) of that
line, to some fixed line in the
same plane passing through the
pole. Let S be the origin or _________
ppole, P the point in the curve,
SP = r, which is called the
radius vector, and Sx the assigned fixed line from which the
angle PSx=-O is measured. If P be also referred to rectangular co-ordinates of which Sx and Sy perpendicular to
Sx are axes, it is easily seen by trigonometry that
rsin 0 = y, rcos 0 = x.
Suppose now that it is desired to determine the sectorial
area included between the radii vectores at two points in a
curve and the arc between them. When a curve is referred
to rectangular co-ordinates x and y, the integralsfyddx or
fxdy between limits determine the area included by a curve
and straight lines parallel to the axes. The relation between
such areas and a sectorial area is established by the following
proposition.
INTEGRAL CALCULUS.
129. Sectorial area in terms of rectangular co-ordinates.
Let PQ in either of the accompanying figures be the curve,
which is taken of such length that it is not met at two
points by any one of its co-ordinates, and PSQ the required
sectorial area.
(1.)
(2.)
Let SK =x, S = X, QK = y, PH= Y. It is evident that
r X
PQKHI = ydx.
Also, triangle QKS = yx, triangle PSH = 1 XY. Also,
Fig. (1), PQS + QSK + QKHP make up the whole PSH;... PQS = -(X - xy) - ydx.
Fig. (2), PQS + PSH makes up the whole figure, as does
also QKHP +t- QSK. Therefore,
- PQS = (XY - xy) f x ydx.
Hence in both cases, PQS, the sectorial area, is, by Art. 34,
equal to
~ jx (fXydxfY
QUADRATURE OF CURVES,
130. Sectorial area expressed by polar co-ordinates. In the
last article the sectorial area was found to be equal to
S(fxdy -./ydx) between proper limits.
Putting x =r cos, y = r sin,
dx = dr cos 0 - r sin OdO,
dy = dr sin 0 + r cos OdO;. d. dy - ydx = r'dO;.. sectorial area = I fr2d0,
where the limits of 0 are the angles between the prime radius
vector and the radii vectores which bound the required area.
131. The same result may be deduced directly from geometrical considerations. Divide the sectorial area by radii
vectores r9, r2, r'... between the extreme radii vectores R, r,
with S as centre, and at distances R, r1, r2... describe circular
S
arcs represented in the figure by dotted lines. The sectorial
area is less than the sum of the sectors of which the arcs are
without it, and less than the sum of the sectors of which the
arcs are within it. The area of a circular sector, of which
the radius is r and the angle S0, is r2 0. Therefore, the
required sectorial area is
less than 1 (R201 + r2SO2 +2 r20, +...) (1.)
greater than I (r21~ + r, 2 + r2A, 1 +...) (.)
INTEGRAL CALCULUS.
where '8,, 02... are the angles between the radii. Now,
r is a finite continuous function of 0. Therefore, by Art. 20,
the above expressions (1) and (2) have the same limit, and as
the sectorial area is between them, it is equal to that limit, or
sectorial area= rd = rdrd, where e,
0 are the inclinations of R, r respectively to the prime radius.
132. Quadrature of the
spiral, r =asinn n, where n
is an integer. This curve has
2 n similar loops, and, therefore, the whole area contained
by it is equal to 2 n times the
area of one loop.
1fr d = - fs2 in 2 0 dO.
Integrating by parts,
J si n nJ. s
"sinn0. sinnOdO=- IcosnO sinnO + os2nOdO
s fe
=- -cosnOsinnO + 1--sin 2nO)do.
Therefore, transposing and dividing by 2, we have
f sin2ndQ = (d 0 - cosnO sinnO ). f.. d a = ( 0 -!cos n sinn )
From the equation to the curve, it is evident that a is the
greatest value which r can have, and that then it is drawn
bisecting one of the loops. Since r = a when n = - rr,
and r = 0 when 0 = 0, the half loop lies between the two
positions of the radius vector corresponding to those values
of 0. Therefore, taking the preceding expression for the
area between limits - and 0 of 0,
2n
area of half loop = a2.
-" n"
QUADRATURE OF CURVES.
The whole area is 4n times this, or = -, which is half
the area of the circle circumscribing the curve. The result
is remarkable, as it is the same whatever the number of
loops of the curve.
133. Of curves, such that one co-ordinate has more than one
value for one value of the other co-ordinate, the quadratures
are found by dividing the curve into several parts, each
of which is of such length that it is not met at two points by
any one of its co-ordinates, and determining by the preceding
methods the quadrature corresponding to each such part.
D
C
B B
0 a Cb d x
For instance, in the accompanying figure the ordinates
parallel to Oy have three values for each value of x between
Oc and Ob, where Cc, Bb, are ordinates touching the curve
at C and B respectively. But the areas AabB, CcbB, CcdD,
may each be found by the preceding methods. Also, the
required area
ABCDda = AabB + bBDd, and bBDd = cCDd - cCBb;... required area = AabB + CcdD - cCBb.
It may easily be seen that the generalization of this rule
is, to divide the area into as many parts as the curve has
parts, alternately receding from and approaching the axis
of y; to find each of these parts by integrating ydx between
corresponding limits; and to take the difference between the
E3
INTEGRAL CALCULUS.
sum of the areas under receding parts of the curve, and the
sum of the remaining areas.
134. Area in terms of the length of the curve. The
parts of the curve which recede from Oy are those for
which x increases as the length of the curve measured
from its extremity nearest to Oy increases; and where,
dx
consequently, if s denote the length of the curve, is
dx.
positive. In the other parts of the curve - is negative.
ds
Now,,/y dx-= p~ dx-.,,
Now, /2dx =f ds (Art. 38).
If, then, sp, so... s, be the respective lengths of the curve
dx
from its commencement up to the points where -- changes
sign, r dx s, dx
Siy- ds, I Y - ds, &c.
gO ds Js1 ds
are the component parts of the required area. But the
alternate parts are to be subtracted from the sum of the
rest. The result will be the algebraical sum of all the parts,
dx
since -d is alternately positive and negative.
ds
Therefore, the required area (S being the whole length
of the curve)
f si dx - s s dx
r= s ds r+ S 2 dxs +......
J ds s d x
+ y ~ds 0= y ds,
+j s ~dsfSJi2?ds
dx
if y d- be a continuous finite function of s. By the nature
as
of the quantities y can only have one value for each value
dx
of s; and, if the curvature be continuous, d has only one
value for each value of s; so that the result of integrating
dx
y- ds is necessarily definite.
13ds. Negative ordinates. In investigating areas of curves,
1.35. Negative ordinates. In investigating areas of curves,
QUADRATURE OF CURVES.
83
it has been assumed that the co-ordinates are positive. When
one of the co-ordinates is negative, the processes described
in the preceding articles will require modification.
By the principles of analytical geometry the symbols +
and - prefixed to symbols of length, are interpreted to
indicate contrary directions of measurement; so that if from
any point in a line curved or straight a length measured
off along the line towards one of its extremities be reckoned
positive, a length measured from any point in the line along
it towards its other extremity is affected by the negative sign.
But no such convention applies to areas which are considered
essentially positive.
If the curve be referred to rectangular co-ordinates, and y
do not change sign between the limits, and x be positive
or negative, fydx is of the same sign as y, if the limits
be taken in the same order as was prescribed (Art. 19)
for positive co-ordinates; that is, if x increase positively in
passing from its value which is the inferior limit to its value
which is the superior limit. This is shewn as follows:fydx is the limit of the sum of terms of the form y.3x,
where ýx, the increment of x, is positive, since x increases
positively in passing from the inferior to the superior
limit; consequently, y Sv has the same sign as y, and fydx
has the same sign.
It follows, that for all areas on the negative side of the
axis of x, fyJdx is negative and Jydx is positive for all
areas on the positive side of the axis of x.
In order, then, to determine the whole area bounded by a
curve, of which part is on the positive and part on the
negative side of the axis of the independent variable, the
two parts must be determined by separate integrations, and
the negative part must be added positively to the positive
part.
136. Negative polar co-ordinates. In determining the sectorial area of curves referred to polar co-ordinates, Jr'dO is
to be taken between limits such that 0 increases positively in
passing from its value at the inferior to its value at the
superior limit. Hence it appears, by similar reasoning to
that used in the last article, that, whether 0 be positive or
negative, fr'-d is positive.
INTEGRAL CALCULUS.
SECTION XII.
CUBATURE OF SOLIDS.
137. LET a solid, ABCcdab, be bounded by a curved surface
abed and by five bounding planes, viz.:-by a rectangle, of
which AB, BC are two sides, and by four planes dA, aB,
Bc, Cd, perpendicular to the plane of the rectangle, passing
through its sides and meeting the curved surface in four
plane curves ab, be, ed, da.
Let the curved surface be referred to rectangular coordinates (x, y, z) of which the axes are parallel to BA, Bb,
BC respectively, and let the surface be such that each
CUBATURE OF SOLIDS.
co-ordinate has but one value for each value of the other
co-ordinates.
Draw within the solid planes, parallel to the bounding
planes and cutting off within the solid, a number of rectangular parallelopipeds, of which, since they are within the
solid, the total content is less than the volume V of the solid.
Add, now, a set of rectangular parallelopipeds (not shewn
in the figure), within which the curved surface wholly lies,
and which are formed by the above-mentioned parallelopipeds
produced. It is clear, that as these additional parallelopipeds
are increased in number and diminished in magnitude, their
sides approach continually closer to the curved surface; and
that, consequently, their volume (v) may be diminished without limit.
V is greater than the solid content of the first set of
parallelopipeds, and less than that solid content + v.
Therefore, V lies between two quantities, of which the
difference may be diminished indefinitely. A fortiori, the
difference between either of them and V may be diminished
indefinitely.
Let the lengths of edges of one of the parallelopipeds be
ax, Sy; z its altitude; z~xSy its volume. Let I z x Ly
denote the sum of the volumes of the parallelopipeds within
the solid V,
V = limit of I z 3x Sy
=jffzdx dy (Art. 117)
= f''d xdydIZ),
the integral being taken between limits which depend on
the boundaries of the solid.
In the figure, for the sake of simplicity, the internal
planes are supposed to be equidistant.
138. The limits of integration for the cubature of a solid
may be investigated by the following method of exhibiting the
result just obtained. Let MM1' NN' be an element of the
curved surface, QQ'RR' its projection on the plane of xy.
Let QQ1 = -x, QR == y. In the limit the solid M'R is
a prism, of which the altitude is z and the area of the
dxdy;.'. dV =- zdxdy.
INTEGRAL CALCULUS.
MI
Suppose the equation to the curved surface gives z =f(x, y).
Then
dV = ff (x, y) dx dy.
In this expression take first (Art. 117) y constant, and
integrate f(x, y) dxdy with respect to x. The result is
the limit of the sum of the prisms, of which the bases are
between the parallel lines q Q', rR'. Let x= X and = x
be co-ordinates of the extremities of their lengths in the
solid;.'. dy dx
a.! X
is the analytical expression of the content of the row of
prisms just defined.
In order to find V, we have to add together this and the
parallel rows of prisms, and to take the limit of their sum.
If Y, y be co-ordinates of the bounding planes parallel
to zx,
V = z dx dy.
139. Solid bounded laterally by a curved surface. We have
in the preceding articles taken the most simple case of
cubature, that in which the solid is bounded laterally by four
CUBATURE OF SOLIDS.
planes. The limits of x and y are then the same for every
point of the solid, and independent of each other. In this
case the integrations are comparatively easily effected. If,
however, the solid be bounded laterally by curved surfaces,
the extreme values of x and y are no longer independent, but
are connected by the equations to these curved surfaces.
Let X, x be constant quantities; Y, y two functions of the
variable x; Z, z two functions of the two variables x and y.
Then it may be shewn that if the volume included between
the six surfaces, of which the equations are respectively
S= X, x = x, y = Y, y = y, z =Z, z = z,
be designated by v,
V = X / Y dxdydz.
From the equations to
the six surfaces it will /
be seen that V is the
volume of a solid, D e, E
bounded by two cylin-...
drical surfaces ECce
and FDdf, of which
the traces are Aa and c
Bb respectively; by
two parallel planes ed, --- --.... _--...
ED, of which AB, ab
are the intersections
with xz, and by two
curved surfaces CDdc A
and EefF. Y
140. Hyperbolic paraboloid. The equation to the surface
of the hyperbolic paraboloid is xy = c z when c is a constant.
The general expression for the volume becomes
S=!ffxy dy dx.
c
Let it be required to find the volume contained by this
surface, the plane xy, and a cylinder of which the base is a
circle of radius r, and the axis parallel to the axis of z.
INTEGRAL CALCULUS.
Integrating first with respect to y between limits Y, y,
V = f(Y2 - y ) x dx.
Now the equation to the cylinder is (x - a)' + (y - b)2= r,
which gives two values of y for each value of x. One of
these values is the superior, and the other the inferior limit
of the integration just performed; or,
Y = b + {r7 - (x - a)21}, y= b - r" - (x - a)}ji;.* Y2 - y = 46{r -. (X - a)};. = 4... V = - {r" - (x - #1} xdx.
The extreme values of x are evidently a +r and a-- r.
Taking the last integral between those limits, it will be
abr2 2r
found that V = r
141. Solids of revolution are those generated
by the revolution of a
plane figure about a fixed
axis. Let the revolution /
of a curve AB about an
axis through A generate
the surface of such a
solid, and let the equation to AB be y =fx,
where x is measured
from A along the axis
of revolution.
It is clear that the
volume of the solid is the limit of the sum of a number of
elementary cylinders having the same axis. Let 1x be the
altitude of one of these cylinders, y the radius of its base;... 7ry2 is the area of the base; and that area multiplied by
the altitude, or vy` x, is the volume of the elementary
cylinder. Therefore, the required volume is equal to
the limit of I(7ry2%x) = rfy2 d x.
CUBATURE OF SOLIDS.
89
142. Content of a cone. A cone is generated by the rotation of a triangle about one of its sides. Let y = ax be the
equation to the straight line generating the conical surface,
where a is the tangent of the angle at which that straight
line is inclined to the axis of revolution. The content of
the cone = c2 Jx2dx = A rs x (taking the integral between limits 0 and x) = I ry2x2, or the solid content of a
cone is one-third the area of the base multiplied by the
altitude = one-third of the content of the cylinder having
the same base and altitude.
143. Paraboloid of revolution. The surface generated by the
revolution of a parabola about its axis, is called a paraboloid
of revolution. To find the solid bounded by such a surface,
and a plane perpendicular to the axis, we must put y2 = ax,
the equation to a parabola.
The required volume ==r afx d -= zr ax2.
144. Solid of revolution through any angle. The quantity
Ir y2dx = 2 ir fJ y dy. Also it is evident, that if the
generating figure turn through an angle q) instead of 2 r, the
solid content generated is equal to
p ff dy dx.
145. Limits of the preceding integrals. If the generating
figure have not for one of its boundaries the axis of revolution, but a curved line, of which the equation is y = Ox,
the limits of integration of ydy are fx and Ox. Similarly,
if it be required to find the solid generated by the portion
of such a figure of which the extreme co-ordinates are two
particular values X and x of x, the integral with respect to
x must be taken between those limits.
146. Content of a solid qf revolution in terms of its area.
Let 5 be some constant quantity. Then if j were equal to the
greatest value of the variable y, ffJ dy dx would obviously
be greater than j.fy dy dx. If f were equal to the least
value of the variable y, ff fdydx would be less than.Jf dy dx. There is, therefore, some value of the con
INTEGRAL CALCULUS.
stant 5 between the greatest and least values of y, for which
ff5 dy dr, or
Sffdy dx = fy dx.
(By Pappus's Theorems, y is shown to be the distance
of the centre of gravity of the generating figure from the
axis of revolution.) The integral on the first side of the
preceding equation expresses the area of the generating
figure. Therefore, from the last article, the content of the
solid of revolution through an angle 9, is equal to
g x area of generating figure,
where g is a line less than the greatest and greater than the
least distances of points in the generating figure from the axis
of revolution.
147. Cubature of a solid of revolution by polar co-ordinates.
Let PSA =0, PS=r be the
co-ordinates of any point P ii
in a plane figure referred to
the pole s. The area of an
element PP' of the figure is
(by Article 131) rd dr. By
the last article, the solid r
generated by the revolution
of PP' about SM through an
angle o, is rdOdr x a distance which is ultimately
equal to the distance of P
from SM, which is equal to a
r cos 0. Therefore, by the
last article, the elementary solid = r cos Od dr, and the
content of a solid of revolution generated by a sectorial area
revolving, about an axis fixed with respect to it, through an
angle 9, is equal to
Sffr cos 0 d 0 dr.
148. Cubature by polar co-ordinates. Every solid may be
generated by the rotation about a fixed axis of a generating
figure of which the form is variable. Suppose the angle of
rotation to be q. Then any solid may be considered to be
generated by the rotation of a figure bounded by a curve of
which the equation is r =f (~, 0).
CUBATURE OF SOLIDS.
When the generating figure has revolved through an
angle 5 + 4, the equation to this curve becomes
r = f (4 + S, 0).
The solid bounded by the two corresponding generating
figures may be always so taken as to be within that generated
by the rotation of one of them, and partly without that generated by the rotation of the other, through an angle ^p.
Hence, ultimately, the required content is equal to that due to
the rotation of either figure; and, therefore, by the last article,
is equal to rpfJP r cosd 0 dr. Hence, the whole required
solid content is equal to.,ffr cos & d 0 dr d.
149. Cubature by polar co-ordinates by direct investigation.
Let an assigned point s be the
pole; let SRQ be an assigned p
plane, and SR an assigned straight
line in that plane. The position
of a point P may be determined
by the length (r) of SP, the radius
vector, 0, the angle at which SP
is inclined to the plane, and q, the Q
angle at which the projection of
SP on the plane is inclined to the
assigned line SR.
(This is evidently similar to oT.
a determination of the distance of s
a point above the earth by its
distance (r) from the observer, its angular elevation above
the horizon (0), and (0) its "bearing" north or south.)
In order to find the solid content bounded by a curved
surface and planes meeting it and passing through the pole
S, suppose that, by a number of planes passing through the
pole, the solid is divided into a number of pyramids having
all their vertices in S.
The required solid content is greater than the sum of
the pyramids within it, and less than the sum of a corresponding set of pyramids partially external to it; and
as the difference between these two sums may be diminished indefinitely, the limit of either of them is the required
solid content.
INTEGRAL CALCULUS.
Let P, P' be two Q/
adjacent points in
the curved surface;
Psp=O, SR =,
co-ordinates of P;
co-ordinates of P'.
Draw through P, P'
respectively, the
planes PQSp and P'Sq'Q', perpendicular to the plane in
which 0 is measured. Also, draw the planes P'QS and
PQ'S, respectively perpendicular to the last-mentioned planes
through P, P'. Therefore the angle PSQ = ýO and
pSq'= =4.
Ultimately, P'S = PS = r, and the pyramid on the rectangular base P'P is an element of the required solid. Now
the content of such a pyramid = - area of base x altitude.
Q'P = q'p = Sp. So ultimately (assuming the proof given
hereafter, that the lengths of a chord and its are are ultimately equal). But pS = r cos 0,.'. PQ' = r cos 90 ý ultimately.
Similarly, QP = rHO ultimately; altitude of the pyramid
=r ultimately;.". its content = r cos 0ý. r O.r ultimately. The required solid content is the limit of the sum
of such elements, and therefore is equal to
ff-1r cos 0 dp dO, or fffr2 cos O dr d dO.
This result is the same of the last article, in which the
same letters evidently signify the same quantities.
RECTIFICATION OF CURVES, ETC.
03
SECTION XIII.
RECTIFICATION OF CURVES AND COMPLANATION OF SURFACES.
AxIoM I. Of lines which join two assigned points, a
straight line is the least.
Axion II. Of superficies which have an assigned plane
perimeter, a plane is the least.
150. Of all lines having the same extremities as a given
curve, and met by planes which meet every point of it but
cannot cut it, the curve itself is the least. This proposition is
proved by an extension of a method given in the Author's
" Manual of the Differential Calculus," Art. 68.
Let AB be the assigned
curve, either plane or of a
double curvature. Then
lines joining A and B and
met by planes which meet E,but cannot cut APB, are all -
of some length, but not all
of the same length. There --
is, therefore, one at least.
of these lines which is the
shortest possible. Let (if
possible) ACB be one of
these lines. Then, by hy- A
pothesis, ACB is met by
the plane at any point P of APB. Two different lines
cannot have common to all their points, planes which meet
but cannot cut them; therefore, the plane through P may be
taken to cut ACB in two points E and F. Therefore, FE,
94
INTEGRAL CALCULUS.
a straight line, is shorter than FCE (Axiom 1). Therefore,
ACB is not the shortest of the lines in question. In the
same way it may be shewn that any other line than APB
is not the shortest, but a shortest exists, therefore APB is
the shortest.
151. Of all surfaces having the same perimeter as a given
surface, and met by planes which meet every point of it but
cannot cut it, the given snuface is the least. Let APB be
the assigned surface,
having an assigned pe- c
rimeter AaBb. Then,
surfaces having that perimeter and met by ------
planes which meet but
cannot cut APB, have '
all some magnitude, but
not all the same magnitude. There is, therefore, one at least of
these surfaces which is A
the least possible. Let.--..-...........
ACB be one of these
surfaces. Then, by hypothesis, ACB is met by the plane through any point P of
APB. Two different surfaces cannot have common tangent
planes at all their points. Therefore, the plane through
P may be taken to cut ACB, which cuts off from that plane
a plane superficies. This plane superficies is less (Axiom
II.) than the curved surface between it and C. Therefore
ACB is not the least of the surfaces in question. In the
same way it may be shewn that no other surface than APB
is the least. But a least surface exists. Therefore APB
is the least surface.
15M. The length of a curve the limit of the length of
a polygon. Let AB be a normal to any curve, CBc (plane
or of double curvature) and Cc a chord intersecting the
normal perpendicularly at D. Draw eBE at right angles
to AB, and in the same plane the normal ACE, and CF
perpendicular to AC. ECF is a right angle;.-. EF > CF.
RECTIFICATION OF CURVES, ETC.
95
Let the arc c BC be of such
length that its curvature is continuous; then F and the curve
are on opposite sides of touching
planes at all points between C
and B. Therefore, by the last
article but one,
BF + CF > CB, but EF > CF; A\.. BE > BC.
Arc CB > chord CB > CD (d
fortiori).
By similar triangles, 0
BE: DC:: AB: AD. e
As the curvature is continuous, the chord Cc ultimately
coincides with the tangent at B, when the arc CB is indefinitely diminished. Hence, ultimately, AD is equal to
the finite line AB, which is the length of two ultimately
intersecting normals, and therefore is a radius of curvature;.'. the limit of the ratio CD: EB is 1. Hence, since the
arc CB is between CD and BE in magnitude, the limit of
its ratio to either of them is 1,
C. B cB
limit - ~ 1; similarly, limit - = 1.
CD cD
CBc arc
Adding, limit -- =h = 1.
CDc chord
Hence it follows, that if in or about any curve of finite
magnitude be described a polygon of any number of sides,
the length of the curve is equal to the limit of their sum
when they are indefinitely diminished in magnitude and
increased in number.
COROLLARY. Let CDc be the arc of a circle of which A
CB
is the centre, and the angle BAC = 0 -- according to
AC
CD
the circular measure of angles. - sin 0;
AC
S CB CB CD 0
S1 = limit limit - -- limit
CD AC AC sin "
INTEGRAL CALCULUS.
Similarly, limit -- 1.
tan 0
153. Rectification of curves. If rectangular co-ordinates,
(x, y, z) and (x + <x, y + Sry, z + az), define two points in
a curve, the distance between them is (Sx2 + y2' + z'2)4,
which is the length of the chord. Hence the length of
the curve is the limit of the sum of quantities of the form of
(ýX2 + y2 + 4)T.
d z" dd z )
+V dy1 d2
1( ~ + dx.
When the curve is plane one co-ordinate may be omitted,
and the expression for the length of the curve becomes
1 + dx2] dx.
154. The superficies of a curved surface is the limit of the
superficies of a polyhedron. Let a polyhedron of any number
of sides be circumscribed about a curved surface which is taken
of such magnitude that its curvature is continuous. Then
all tangent planes of the curved surface cut the polyhedron.
Therefore (Art. 151), it is greater than the curved surface.
Within the curved surface inscribe a similar and similarly
situated polyhedron. It is clear that planes may be drawn
through every point of this polyhedron, which do not cut
it, but cut the curved surface. Therefore, by the same
article, this polyhedron is less than the curved surface.
Also, in a continuous curved surface, an inscribed plane
ultimately coincides with a tangent plane when the surface
subtended is indefinitely diminished. Therefore, the edges
of the inscribed and circumscribed polygons ultimately coincide, and the limit of the ratio of the lengths of two homologous edges is 1 (Art. 152).
Also, their homologous sides, being in the duplicate ratio
of their homologous edges, have 1 for the limit of their
ratio. Therefore, the surfaces of the polyhedrons are ultimately equal. Consequently, the curved surface between
them is ultimately equal to that of either polyhedron.
155. Section of a parallelopiped. The following proposition
will be required in determining the complanation of solids.
RECTIFICATION OF CURVES, ETC.
97
Let ABCD be the base of a rectangular parallelopiped, of
which the sides AaD, aB, bC, cCD are cut by the plane
abcD, which is a parallelogram. Its area is required.
z
y
C)V~
D
In the right-angled triangle a AD, aD2 = Aa+ A D', ('.)
Similarly, Dc2 = DC2 + Cc2, (2.) To find the distance a c,
let a perpendicular cc be drawn from c on to Aa. Then
ae = Aa - Cc, and in the right-angled triangle ace,
ac2 = ce2 + (Aa - Cc)2 = A C2 + (Aa - Cc)Y
= AD2 + CD + (Aa - Cc)', (3.)
In the triangle aDc, by a trigonometrical formula,
ac2 = aD2 + cD - 2aD. cD cos aDC; or from (1), (2), (3),
AD' + CD' + (Aa - Cc)2 = aA2 + AD + DC2 + Cc2
- 2 (aA' + AD') (Dc' + Cc)I cos aDC;.. Aa. Cc = (aA2 + AD ) (DC- + Cc2)B cos aDC;
also required area abcD = aD. cD sin aDc, and
sin2 aDc = 1 - cos2 aDc;.. (abcD) =
(nl~n ~(o~c{C Aa'. Cc'
(aA + AD2)(DC2 + Cc2) f I ( A a2. Cc2
(a A + A D-) (DC2 + cC)
abcD = (aA2. DC2 + AD2. DC2 + ADW. Cc?)i.
156. Complanation of suifaces. Let the surface be referred to rectangular co-ordinates x, y, z. Also, suppose the
surface be cut by several planes parallel to the planes xz, yz,
respectively. Then, by Art. 154, the surface is equal to the
INTEGRAL CALCULUS.
limit of the sum of the sides of an inscribed polygon, and
therefore is equal to the limit of the sum of parallelograms
inscribed within the surface and bounded by the supposed
planes.
In the last figure, let AD be parallel to the axis of x;
AB to that of y; Aa to that of z; and let (x, y, z) be the
co-ordinates of D and DA = 8x; AB = ay. Also let D, a,
and b be three points in a curved surface. Then, if in the
equation to the surface, when x is increased by <x, and y
does not increase, z be increased by xz, Aa -= xz. Similarly, if Sz be an increment of z, due to an increment
)y, x not increasing, Cc = yz. Therefore, by the last
article,
abcD = (8 '_. a + z 2. + 8'.2.
Hence the required surface is equal to the limit of the
sum of terms of the form
( + -~-* + - a
or the surface
= xd I + 1z 2 i ) ddez ( 2
where the parentheses indicate partial differential coefficients.
INTEGRATION OF DISCONTINUOUS FUNCTIONS.
SECTION XIV.
INTEGRATION OF DISCONTINUOUS FUNCTIONS.
157. THE Definitions of Integrals, Arts. 17 and 115, were
restricted to finite continuous functions of a finite variable,
and the principles of integration were established on the
tacit assumption that the integrals were finite exact quantities,
and that, consequently, each function integrated had a single
determinate value for each value of its independent variable.
If, therefore, a function be discontinuous, or have infinite
or indeterminate values between the limits assigned for integration, or if either of these limits be infinite, the preceding
definitions do not apply to it. It may be observed, that the
accuracy of most of the foregoing theorems depends essentially on their application to finite functions, and is violated
by the violation of this condition.
158. The following is an instance of the errors that would
arise from application
of the theorems of the y
preceding sections in
neglect of the consideration of the last paragraph.
Let =- be the
equation to a curve re- p
ferred to O x, Oy, as
rectangular axes. These - o - A.
axes are asymptotes of
the curve, which has two similar branches.
The area included by any portion of the curve, the ordinates at its extremities, and the axis of x, is equal to
fydx between corresponding limits (Art. 19), if the function integrated be finite and continuous between those limits.
Therefore, the area
Sa dx 1 1
APQb JI =- ---,
F " ':"1
INTEGRAL CALCULUS.
if OA = a, Ob = 6. This value of the area is increased
indefinitely as b is diminished. We may, therefore, make
the area APQb as large as we please by taking the point
b near enough to 0.
If, however, we integrate from a to - a, we find the area
APpa = -
if Oa =- a. And this result is evidently erroneous, for it
gives the expression for the area, which ought to be positive
(Art. 115), a negative sign, and it makes it equal to a finite
quantity; whereas it has been proved, that of the area a
portion may be taken indefinitely large. The error arises
from integration through an infinite value of the integrated
function.
159. The meaning, then, to be assigned to integrals of
functions which are infinite or discontinuous between the
limits of integration, is up to this place purely arbitrary; a
definition of such integrals may, however, be given, which is
so strictly analogous to the preceding definitions, as to render
obvious theAethods of extending to discontinuous functions
the principles already demonstrated.
DEFINITION. If fx become infinite, impossible, or discontinuous for either or both the values x=a, x= b, but
not for intermediate values, let fxdx be defined to be
the limit of fx dx, when, and a, are any continu+ 3-,
ous quantities which have the limit zero; a - 8, and b + a,
being values of x, between a and b.
More generally, if fx become infinite, impossible, or
discontinuous for the finite number of values a, b, c... m,
and for none else, of x between X and x, let, by analogy
r X
with Art. 27, fx dx be defined to be the limit of
J x
r X p a-, - I bf xdx+ J fxdx + + "fxdx +...
a ++ J dx. (a),+
+ xdx "/ ^... (a),. * * ^ X
INTEGIRATION OF DISCONTINUOUS FUNCTIONS.
101
when a,, ',... are any continuous quantities which have
the limit zero; a - 8' and b + J2 being between a and b,
b - ',2 and c + a, between b and c, &c.
SX
160. Principal values of integrals. The value of X f xdx,
x/ X
as just defined, may be dependent on the relative magnitudes of the arbitrary quantities,, ',... If these quantities
be assumed to be all equal, the integral has then what is
termed by M. Cauchy its principal value.
EXAMPLE.-The following is an instance of an integral,
of which the value, according to the above definition, is
essentially arbitrary:S+a dx. a dx ~-, dx
- = limit ( -
J -+a X X -a X
= limit ( l+ /2 Art. 39, IV.
= limit log, = log, limit, (Art. 15,)
a quantity to which any value whatever may be assigned at
pleasure, by assigning a corresponding relation between the
arbitrary quantities 8,, 2,.
If in the preceding result = -, we have the "principal"
value of the integral equal to log, 1 = 0.
161. Condition that integrals may be determinate. Every
function which is finite and continuous between any exact
limits, either continually increases or continually decreases,
or alternately increases and decreases an exact number of
alternations. Take two limits, between which it continually
increases or decreases. The integral of the function between
those limits is (Art. 22) between its two finite quadratures,
and is, therefore, a finite quantity. It is also determinate,
not arbitrary, for the only arbitrary quantities in the quadratures disappear from them in the limit, Art. 26. Also,
the whole integral between any finite limits is the sum of
integrals, such as that just considered, and of which the
INTEGRAL CALCULUS.
number is that of the alternations referred to. Therefore,
the whole integral is an exact quantity.
If, however, the function to be integrated be not always
finite and continuous between the limits of integration, the
integral is the limit of the sum of the integrals of (a) in
the last article but one. If the limit of all of them be
finite, f fxdx (their sum) is finite. It is then also
x/ X
determinate. For each of the integrals of (a) is determinate
according to the last paragraph, and the only arbitrary quantities 68, 8'... disappear in the limit.
Hence, when / fxdx is either infinite or indeterminate,
the integrals in (a) have not all finite limiting values. If those
SX
which are infinite in the limit be all positive, / fxdx
is evidently equal to + co; if they be all negative, to - oo.
Hence, the only case in which fxdx can be indeterminate or arbitrary, is when more than one of the integrals in (a) are infinite, and have different signs in the limit,
X
when / fxdx takes the indeterminate form (adding together the infinite quantities with like signs) co - o.
f a dx
For instance, in the last example, / - is the limit
J -- X
of the sum of two integrals, of which the first has the limiting value + co, and the second -- c.
162. The preceding principles may be illustrated geometrically. First, with respect to finite continuous functions:
let y be such a function of x, and x, /, the co-ordinates of a
plane curve which will be unbroken, since the function is
continuous. Whatever may be the form of the curve, a finite
area is included by a finite portion of the axis of x, the
ordinates at the extremities of that portion, and the arc
between them. But this area is equal tofydx, taken between finite limits.
Next, let the function be not always finite and continuous.
Then it will be represented by a curve, y =fx, which has
infinite branches, or breaks, or both.
INTEGRATION OF DISCONTINUOUS FUNCTIONS.
Where there are breaks only,
as from B to C and D to F, and y
not infinite branches, let a and b E
be the values of x at the points
a and b in the diagram. Then r D
the area aABb is evidently equal C
t i f!b-?1
to the limit of + ý2 ydx, a o b 6
finite quantity. Similarly, the
areas bounded by the other parts
of the curve are expressed by the limits of integrals of the
form of those in (a), Art. 159; and the quantityj fxdx
in that article represents the whole area of the curve, which
is equivalent to the sum of the areas of its parts.
If the curve be of the form
AB, CD, and have no values of y
between Bb, Cc, the function is y
impossible for the infinite num- c
ber of values of x greater than B
Ob and less than Oc. Then the A
definition of Art. 159, which is
restricted to functions with a finite
number of impossible values, is
inapplicable. In order to inter- 0 - 0
pret geometrically or analytically
integrals of such functions, another definition would be required, as essentially arbitrary as that just mentioned.
Next, let the curve have infinite ordinates y for finite
values of x. These ordinates are asymptotes of the curve,
and the area bounded by the infinite branches of the curve
may be finite, as in instances given in Arts. iMO and 127.
If ordinates y be all positive, these areas are positive,
and their sum is the quantity f f xdx, which is now
under consideration. If some of the ordinates be negative,
the corresponding areas are negative (Art. 135), and the limit
of some of the integrals in (a), Art. 159, will be negative;
so thatf/ f xdx, the algebraical sum of the limits of those
integrals, will represent the difference between the total
areas on opposite sides of the axis of x.
INTEGRAL CALCULUS.
fx
Lastly, let the curve be such as to represent / fxdx
in the form o - o. The
curve, of which the equation y
1
is y= --, has two similar infinite branches; one on the
positive and one on the negative sides of both axes, which
are asymptotes. Let OA=a, A, ',
OB=~1. The area BbaA - o A
radx a a.
---=log
b
Let OA' = - a,
OB' = - -;
area BW'b'' d / (Art. 135)
f -Ba X
S (Art. 39, IV.) =log -
6- a x a
m d. -..t. ad -~dx
The integral - d is the limit of - + -
_ -a X f t: x -_a, X
= limit of (area BbaA - area B'b'aA') as B and B' approach O. But the difference between these two is arbitrary,
for it depends on the ratio of the two arbitrary quantities
OB, OB'. If we choose to assume OB = OB', the two areas
B baA and B'b'a'A' are always equal; their difference is
then zero, which is, therefore, the "principal" value of the
integral -.
J a dX
163. Integrals with infinite limits. The definitions of
integrals (Arts. 17 and 159) were restricted to finite limits.
The extension of the definition to integrals with infinite
limits, may, by obvious analogy with preceding cases, be taken
to be the limit which the integral with finite limits approaches
when either or both limits are indefinitely increased.
INTEGRATION OF DISCONTINUOUS FUNCTIONS.
105
164. Multiple integrals of discontinuous functions. Many
of the principles, of this section respecting integrals of one
independent variable may be extended to multiple integrals.
For instance, it was shewn in Art. 118, that the result of
multiple integration of finite continuous functions is the
same in whatever order the several integrations be performed. This principle does not hold for functions which for
particular values of the independent variables between the
limits of integration become infinite.
oy -- x1
For example, (X +, if x first approach the limit 0
and then y, has the limit oo; and, if y first approach the
limit 0 and then x, has the limit - co. We cannot, therefore, affirm, that
Sdx dy and
t* b ( X + Y2),
p dy / dx X,
have the same result.
J (x + Y) d + y' x + b"'
taking the integral between limits, y = b and y = - b,
0, b b ta n- - = - 4 tan-1 i
Jx 2+ b b b'
taking the integral between limits, x = a and x = - a.
Now reverse the order of integrations.
-_- x, x 2a
=-- d - d a =
+ - y2 + y' +I a+
Sr -dyJ =_ 2 tan-1 = 4 tan-1 b
J y + a a a
F 3
INTEGRAL CALCULUS.
taking the integral between the same limits as before.
Hence the two results differ by
4 tan- + 4 tan-1~ =4 - - tan- +4 tan--= 7r.
Sa \2 a a
165. In order that multiple integrals of discontinuous
functions may be the subjects of exact investigation, a new
arbitrary definition is requisite. The following is an obvious
extension of the definition for discontinuous functions of one
variable.
DEFINITION.-Omit ranges of values of the function between arbitrary limits which include the discontinuous values.
Integrate the function for the rest of its values. The limit
of the result when the ranges of excluded values are as far
as possible contracted is the required integral.
166. To illustrate the definition, suppose, first, that there
are only two independent variables, x and y. Consider them
to be rectangular co-ordinates of a point, of which f(x, /),
or z, is the third rectangular co-ordinate. Then z =f (x, y)
is the equation to a surface. Suppose, first, z to become
infinite only when drawn from an isolated point (a, b), in the
plane of x, y.
Now, inclose the isolated point by any contour in that
plane. Then integrate for all values of z drawn from points
in the plane of x, y, without this contour. The result is, the
volume of the solid under the supposed surface, minus the
content of a tube surrounding the infinite ordinate. The
analogy with the preceding definition requires that the bore
of the tube be diminished indefinitely. Now, the bore or
contour may diminish an infinite number of ways. Its
ultimate form may be any curve or a point.
Again, all things else remaining as before, let z be infinite
when drawn from any point of some finite curve in the plane
x, y. Surround this curve by a contour on the same plane.
The solid, minus the content of the tube, having this contour
for its bore, is taken as before; but in this case the contour
necessarily contracts into the assigned curve.
167. If the function include three independent variables
x, y, z, we may regard f (x, y, z) as some kind of magnitude
(a mechanical magnitude, for instance,) which depends on
INTEGRATION OF DISCONTINUOUS FUNCTIONS.
the position of points in space. Then, without assigning a
meaning for the integral, we may suppose that the function
becomes infinite, either at an isolated point, or at all points in
a certain line, or all in a certain surface, or all in a certain
solid. In either case, suppose the point or points surrounded
by a surface. The required integral is the limit of that
of the remaining solid when the surrounding surface is contracted to the utmost. When its ultimate form is a surface,
the equation to it gives one relation between the variable
limiting values of x, y, z; when the ultimate form is a
line, the equations to it give two relations; when the ultimate form is a point, three. In the same way with n
independent variables, it may be conceived that 1, or 2,
or 3... or n such relations exist, of which, some may be
arbitrary.
168. The required integral, consequently, may depend
on arbitrary relations, and itself, therefore, be arbitrary.
Where, however, the function is such as to be infinite only
for isolated values of the variables, and is the same in whatever manner the ranges of the excluded values are contracted, the following method gives the required determinate
result.
Let a function f(z, y, x... s, r) become infinite or discontinuous for a finite number of values of the independent
variables of which those of r are aa, a2, a,... a,,,, and none
else between R and r. Also, let the required integral
d. dy... dr f (z, y... r)
be reduced (Art. 117) to the form F (r) dr, by the
successive integration of f(z, y... r), and other functions
(which have not discontinuous or infinite values until a, a',
a,... be substituted in them for r). Then the required
integral may be considered to be the limit of
F (r) dr F (r) dr + F (r)dr +...
+a, + 2 ' F (r)dr,
+ ^aln 111 F (r) dr,
t^?
INTEGRAL CALCULUS.
when al, al'... are any continuous quantities which have
the limit zero; a1 -- and a2 + a8 being between a1 and
a2, a2 - a2' and a3 + 6, between a and a3, &c.
169. The integral is independent of the order of integration.
Let s designate the independent variable preceding r in the
order of integration of f(z, y... s, r), so that
sf (r,) ds = Fr,
just referred to. The integral is, by the preceding suppoition, the limit of
J ddr f (r, ) ds + a,- d f(r, ) d+...
+ " ' d r f(r, s)ds......... (1.)
Let, b2...,, be the values of s, which correspond to
ai, a2... a,, of r, to render the original function discontinuous
or infinite. It is required to shew that (when el, 6'... have
the limit zero) the limit of
Sds f (r, s) dr +, ds f (r, s) dr +...
+ ' "L4 ds f (., s) dr...... (2,)
is the same as that of (1), if that be not arbitrary.
For brevity, omit all the symbols of integration except the
S
limits. Then indicates the operation of integration of
f (r, s) between limits S and s. Then, since f (r, s) is a
continuous function, while the value of r is general,
S S b,+6 b,-',1 b2+s, b +s b -s'
8 blS +s b,-s + b2+E2 b2-s " - + 8
by Art. 27. Therefore (1) becomes, supposing the operation
written outside each bracket to be performed on all within it;
INTEGRATION OF DISCONTINUOUS FUNCTIONS.
109
R S 5+ b -$/I,,+i. + -i
Sl + s- -i- b+'1 bm--+ s +,
a+ - +1 + 1- 1 +- +. -.b2 m b }
S,+'+, I b,- + 1.., + bni+, (i.)
+a- l+ bl--ts l -- b2+2 bi-l ' l
+ &c.
al-, S bl+s, bl-1 b,, + b,, -'
+r fbi+s + + + 8b a+
r bl"i4 - - i '1 62+ 2 $'2 b -,n tm 1
In the same way (2) becomes
b1+- 1 ai +ai+ a-a1 + aa r J
b -- -R a Y+,l a,-l ' al+ ~ a -- -
+, + + a2+, + + a'+ + ++ r " (II.)
+ &c.
s - 1+ { 1 4-,+, a+,-', a,+4-m,-,
+s 8 a,+ + -, -/I+ a+; +' " + +-a r
It will be found that the alternate expressions, beginning
with the first and ending with the last in the { }, correspond to integrals which are common to (I.) and (II.).
Hence, the difference (I.) - (II.) does not contain those
integrals.
Of all the remaining integrals, the limits written in the
{ } indefinitely approach each other when El, E'... 1, 8 '...
approach zero. Hence, the limit of each of these integrals
is zero. Consequently, as their number is finite, the limit
of the difference (I.) - (II.) is zero. Therefore, (1) and (2)
have the same limit. This result shews that it is immaterial
with respect to which independent variable the final integration is performed. And, with respect to all the other
independent variables, the order of integration is proved to
be independent in Art. 117.
INTEGRAL CALCULUS.
SECTION XV.
DEFINITE INTEGRALS.
170. THERE are many functions, as has been already
stated (Art. 40), of which the indefinite integral cannot be
expressed in finite terms by ordinary algebraical, logarithmic,
and circular functions; where, however, general integrals
cannot be found, integrals between particular limits may be
frequently determined. For instance, a e- dx cannot
be expressed by a finite number of algebraical or trigonometrical functions of a and b; but
0 oo 2
Se- x = r,d
as will be presently shewn.
The subject of definite integration is of great importance
in difficult mathematical investigations, and it frequently
happens that the particular limits between which definite
integrals can be most readily determined, are those to which
such investigations lead. The scope of this treatise will not
allow of more than a very brief notice of one or two of the
most important principles of definite integration.
o1 1 \n-1
171. The second Eulerian integral.f (log, -- dz,
which is equivalent to x'- e-"Xdx when log- =,
derives its name from Euler, who first investigated it. It is
designated by Legendre by the symbol r(n), where n is
positive. The integral is evidently a function of n only.
172. To determine f d x"E- dx, where n is a positive
integer. In Art. 80, write P = e-";.'. P, - a-1 e-a
2--= a-2 e-ax, &c. Therefore,
DEFINITE INTEGRALS.
J e-' C"'dx = e-"ax (a-' " + a-2. nx+ a--3. n.n-1.x'"-2 +... + a-0b+1.n. n-1...2.1).
When a becomes infinite,,e -,x has the limiting value
zero, by evaluation according to the methods of the Differential Calculus;
" e-ax d a- n+) 1.. 3...n. When a =,
7f e-xdx = 1..... n = (n + 1)
0 co
by the last article; r(2)=1; r(3)= 1.2; r (4)=1..3, &c.;
12. 2.39...p = [r (p + 1)]2.
173. To investigate 0"e0-ax dc, when nit is nzot an
integer. Changing x into ax in the equation
/3 oo
d an-e- dx= r (n), we have
'0 Xn-1 6-ai d (.... (a),
fo ~ al
for all positive values of n. Integrating by parts,
fe- X" dz =- e-" ' + n fe-x 1"- dx.
Taking this between limits a = cc and x = 0, we have
r (n + 1) = r n for all finite positive values of n. Similarly,
r (n + )=(n + 1) r (n + 1), r (n + 3)=(n + 2) r (n + 2), &c.
174. The first Euleriana integral. TIn (a) Art. 173, wvrite
p + q for n, and 1 + y for a. Then
00 r ( p + q)
SP+ql-1 ( -(lI+y)X( d= (
Multiplying by y q-1dy, and integrating between limits
co and 0,
1NTEGRAL CALCULUS.
f f/ 0 xp+q- yq-1 e -(l+.v) dy dx
= (p + q) 0 y-1 y
-r( (l + y)P+q
The multiple integral may be integrated first with respect
to y, considering x constant (Art. 117). The resulting
integral is similar to that of (a) Art. 173. Hence, the
multiple integral becomes
o e- P+q-Idx = r q xe-i-Zdx = I'q. rp.
Whence from the preceding equation,
rp.rgq o Oq y-ldy
S(p + q) Io ( + y)P+ q
The integral is called the first Eulerian integral, and is
designated by the symbol (p I q), by Cournot. The preceding formula is the fundamental relation between the two
Eulerian integrals, It is evident from it that
(pIq) = (q Ip).
175. Ultimate ratios of Eulerian integrals. In the first Eulerian integral put 1 + y = e. Then, when y = 0, z = 0;
and when y = o-, z = co; so that the limits of the integral
are not changed. Also, dy = - e' dz, and the integral
P
becomes
p. (p _- 1)q-1 Epdz r 0c (E - _)q-1dz
0 z (p + q) f z (q -1)
=p-q f {p(l - PP)}q- -lE-dz.
All the steps by which this result is obtained hold when
p is indefinitely increased. Then the quantity in the { }
0 and by evaluation by differntiamay be put in the form 0, and by evaluation by differentia
DEFINITE INTEGRALS.
113
tion becomes z. Hence, when p is indefinitely increased,
the first Eulerian integral
(p 1 q) becomes p- 00 z q- e-z dz = -p.
pq
Therefore, substituting in the last article for (p 1 q),
rp 1 r (p + q)
r (p + q) pQ rp
when p is indefinitely increased.
If in the last result we put for q, successively, 1 + n and
1 - n, and multiply together the results so obtained, we have
r(1 p + 1 +n).r(p+ 1 -( n)
p' [rp]2
r(p + n).r(p + 1 - 1 n)
[r (p + 1)]f
(Art. 173), when p is indefinitely increased.
176. Multiplying together a series of the equations at the
end of Art. 173, p + 1 in number, and omitting common
factors,
r (n +p + 1)
n.n+l.n+ 2...n+p.
r (n)
r(p + l-n).. 1-n.2--n. 3-n... p-n. ==
r(l - n)
writing 1 - n for n, and p - 1 for p. Multiplying together
these two equations, we have
12-n2. 22-n2. 32-n2... p -n2
r(p + n + )r(p-n +1)
S nrnr(1 - n)
n,. 32 n.
r(p+ n+ 1) r(p-n+1) 1
l2.22. 3ý... p2 nr(n)r(1-n)
r(p+n+1).r(p-n+ 1) 1
[r (p + 1)]. n r(n) r ( - n)1
Er < [(+lJ.1r>^-^
INTEGRAL CALCULUS.
By Art. 175 the first fraction on the second side of
this equation converges to the value 1, as p is indefinitely
increased;
2 2 2.
%1.a1 1if.
12 22 3" i (n) 1 (1 - n)
sinnir 1 iT
sinr or r (n)r (1- n)
nwr no (n) r(1 - n) sin n2
IHence when n=,
(_I)]2 -2, v
[r 7r, r Q) = 7r- dx.
L"?/1 1- c-1 I xAlso, writing 1 for n, r( ) r )= r7
sin -
21~ 7rfor n, r( -r 7r
sin -
for n, - 1
a2 n?TE. - 1
sin1 --- 7
Multiplying (n - 1) of these equations together, and reS. r. n-1 n
membering that sin-.sin--... sin - 7 = we
n n n
have
[F(... r]( -7
From x- e-dx =?r, we easily find
<0
00
E-x2 dx = - r0, putting x2 = x.
e0
177. To investigate J dv e-ax cos ra. Integrating by
parts,
DEFINITE 1NTEGRALS.
fdx c" cos rm = - e-"x cos r - - e-" sin rx dx
r O 1. COS r
Sx e-"ax sin rx = - -e"a sin rx + J e cos rd;
rd a cos rx - r sin rx.'. x e-a cos rx = - e --+ r
J a2 4- r"
x - - a sin rx -- r cos rx
dx e sin rx = - a2 + 2
These integrals are to be taken between limits x = O and
x = 0. When a is positive and not zero, e-"a is zero at the
former limit, at which also the fractions on the second sides
of these equations are finite if a and r be not zero, since
sines and cosines are finite by their definition. Again, when
x has the limit 0, e-"x = 1 if a be finite; the numerators
of the fractions become a and r respectively, if a and r be
finite. Hence
dx e-"ax cos rx =
J a + rJr0 a + r 2'
od x s-a' sin r = -......a(1.)
178. Sine and cosine of an infinite angle. If, in defiance
of the restrictions with respect to a and r, by which these
results are obtained, we put a = 0, r remaining finite, and
assume that e -o. = 1, for all values of x between its
limits, the results apparently become
0fo 1
dx cos rx -= 0; dx sin = -...... (2,)
whence, since
fd sinrx rd cosr?
fdx cos r x --, dx si rx Co- -OS
r r '
it would follow that cos oo = 0 and sin oo = 0.
But it is essential to the evaluation of the original definite
integral that a x = w, when x = oc; a condition which re
INTEGRAL CALCULUS.
quires an arbitrary relation between x and a if the latter
have the limit 0. Moreover, the supposed values of cos o
and sin ow violate the relation sin2 + cos2 = 1, which is part
of the very definition of "sine" and "cosine."
The antecedent objection to assigning a definite value to
the sine or cosine of an infinite angle is perfectly insuperable; for, however great a number of times the radius
describing the angle revolve, the sine and cosine will vary
from 1 to - 1 in the course of each revolution.
The correct statement to be substituted for equations (2)
appears to be, that the original definite integrals of e-" cos rx
1
and E-a sin rx, approach the limits 0 and - respectively,
r
when a approaches the limit 0, r remaining finite.
Since equations (1) are true for all finite positive values
of a and r, let r2 = n a where n is any arbitrary number.
Then, the first equation of (1) becomes
/ 0 1
Sdx e- axcos (na) x = ---=
j 0 a + n
If it were allowable to put a = 0, we should have in strict
r 0 1 1
analogy with (2), dx -.. = -, any finite arbiJ0 i n
trary quantity, - a result which obviously contradicts the
fundamental principles of the Integral Calculus.
/a Co
179. To investigateJ dx e-lx cos2 x. By integration
by parts twice, it is easily found that
SC -" sinx - acosx 2 e-,x
J ao + 4 a (a + 4)
When x = o, e-a is zero for all positive values of a not
zero, and therefore the second side of the preceding equation vanishes. When x = 0, the same side becomes
a 2
ad + 4 a (a2 + 4)'
oo a +- 2+.". E"" C os2 dx ==
0 a (a" + 4)
DEFINITE INTEGRALS.
117
180. Differentiation of definite integrals. The differential
coefficient with respect to c of a definite integral
/' a
b f (x, c) dx,
is found by differentiating under the fthe function f(x, c).
Let F be the integral, and F its increment, due to an
increment 6c of c; and let f (x, c) be the corresponding
increment of f (x, c).
S=f f (x2 c + a c) dx - a(, c) dx
=- a {f(, c + c) -f(, c)} dz,
af a af (x, c) d af d f (x, c)
j f dx, and / dx,
ac b ac dc A db
when 8c has the limit zero.
-k 00
181. To investigate dx e-a'x cos fex. The prin< 0
ciple of the last article is remarkably illustrated by this
integral. Calling it F,
dF ro
d =- dpx e-ax' sin 2 c...... (1)
= a-2. e-a2 sin 2cx)~-ca-2 dxd: -aS" cos 2cx,
integrating by parts. The quantity in the bracket disappears
when taken between the assigned limits, for all finite values
of c, a not being zero;
dF dF. - 2ca;.F. 2 -- ca.dc.
dc F
Integrating, log, F = - ea-2 + a constant, or F = C e-c2a-2.
Equation (1) and all that follow from it are true for all finite
values of c, positive or negative. Therefore, if in the last
equation, c having the limiting value 0, we have
118 INTEGRAL CALGULUS.
C d= e-a2X2 z- e-icz dz,
f 2a 0
gri
putting a2x2 = z. Hence, by Art. 176, C =
2a'
cIX EI2~ COS ~-"~ CXs =2 - E-a.
1 00 9r -c2a-2
This integral is due to Laplace:-Memnoires de l'Institut,
1810.
APPENDIX.
DEMONSTRATION OF TAYLOR'S THEOREM.
LET any function (f) of a single variable and its successive differential coefficients (f', f", &c.) be finite and
continuous for all values of the variable from a to a + h.
In the expression
x2 N - 1 xn
S( +x) -f a-f 'a. x-. - --- - r...a(1),
1.2 1.2...n-1 1.2...n
let R be such a finite quantity, not involving x, that when
x = h the expression = 0. It is also zero when x = 0.
But a function which is zero for two different values of its
variable cannot be always increasing nor always decreasing
in the interval. Hence there is some value (x,) of x between 0 and h, for which the differential coefficient of (1)
(i. e. its rate of increase) is zero; or,
f' (+x)-f'a -f"x-f" - -...-. X... (2),
1. " 1.2...s - 1
is zero when x = xi; (2) is zero also when x = 0. Therefore, as before, there is a value of x between x, and 0,
for which the differential coefficient of (2) is zero. Continuing the process to n differentiations, we have, finally,
f"(a + x) - R = 0, when x has some value between 0 and h.
Let this value be 0h where 0 is a proper fraction. Then
R =f "(a + h), Substituting this value of R in (1), and
putting (1) = 0 when x = h,
h" h"
f(a+h)=f(a)+f'a. h +f"a --.. 2 + f +f(a + 0h)-..
which is Lagrange's Theorem on the Limits of Taylor's
Theorem.
120 INTEGRAL CALCULUS.
If the last term of this series become zero when n is
sufficiently large,
h2
f (a + h) ==fa +f' a. h + f" a. h +... to convergence,
which is Taylor's Theorem.
This demonstration is a somewhat simplified form of one
originally published by the Author, in the "Cambridge and
Dublin Mathematical Journal," vol. vi., p. 80, and reprinted
in his " Manual of the Differential Calculus," Art. 54.
2. TAYLOR'S THEOREM DEMIONSTRATED BY INTEGRATION.
By successive integration by parts,
Jf'(a + h- )dz=zf((a+h -z) + Z f"(a + h - z)dz
=zf'(a+h-z) + - P~(f+h-z) +, - f"(a+h- z)dz
Z3
= zf' (a~ + 7- z) + -f"(ai-z) ~ f"'(a+7-,a)+
1.2 1.2.3
+ ffn 2 1f(a + h- z)dz.
Take this result between z = h and z =0. The first side
of the equation becomes, by Art. 39, (III.), f(a + h)-fa.
Then, transferring fa to the second side of the equation
taken between limits,
h=
f(a + h) =fa +f'a.h +f"a. +...
+. n f-(a + -z)dzg
which expresses the remainder of Taylor's series by a definite
integral.
G. Woodfall and Son, Printers, Angel Court, Skinner Street, London.
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