Thus the problem requires two processes of integration; (1) we must integrate dy, or, as we say, find the y-integral between the limits 0 and f(x), the former value being the inferior, and the latter the superior limit of y; the result of this integration is evidently f(x); (2) we must integrate f (x) dx, or, as we say, find the x-integral, from x = x, to x = x; the result of this double definite integration will give the area Mo Po P, M, in terms of o and x

n

It will be observed that in the last example the infinitesimal element is an infinitesimal of the second order; that after the y-integration it is an infinitesimal of the first order; and that the final integral, which expresses the area, is finite. This is in exact accordance with the geometrical quantities expressed by the several symbols.

4.] I return now to the general problem of integration of functions of one variable as expressed in equation (1); and to the creation of a convenient system of symbols.

Let s as heretofore represent the definite integral of f(x) dx between the limits x and x; and let x-xo be divided into n infinitesimal parts; and let x1, x2,...,_, correspond to the (n−1) points of partition; then by (1),

2-1

S = f(x)(x1−x ̧) +ƒ (X1) (X2−X1) + ... +ƒ (xn−1) (xn− xn−1). (4) Now on referring to Vol. I, Art. 8, and the mathematical definition of a derived function there given, it appears that if F(x) is a function of z whose derived function is f(x), and x, and x, are X1 xo two values of a differing by an infinitesimal, F(x) and f(x) being finite and continuous for all the employed values of their subjectvariables, F(x1)—F(x) = ƒ (xo) (X1−x。)·

x

(5)

Similarly, if F(x) and ƒ (x) are finite and continuous for all the values of their subject-variables employed in the following equations, F (X2) —F (x1) = ƒ (X1)(X2—X1),

F(x) — F(X,_1) = ƒ (Xxn_1) (X„— Xn_1

(6)

Let all the right-hand members and all the left-hand members of these several equations be added; then by (4) we have

We require a symbol to express the relation of r to ƒ; as d or D expresses the differential or infinitesimal element of a variable or of a function; so we employ (a long 8) to denote the general

sum of an infinite number of terms, each of which is an infinitesimal. Thus if ƒ (x) dx is the type-element, the sum of an infinite number of which is to be determined, that sum is represented by

[f(x)dæ. Also as thus far the limits of integration are not in

troduced, this symbol is used to represent the indefinite integral. And thus as d or D is the symbol of differentiation, so is symbol of integration.

is

ft

the

The definite integral is conveniently expressed as follows: If Xn , and are the limits of the Integral, x, being the last or the superior, and the first or the inferior limit, then these symbols

may be placed at the top and the bottom of respectively; so that the definite integral thus determined is expressed by the symbol

[** ƒ (x) dx.

Also since F(x) is that function whose derived function is f(x), let us represent, as in the Differential Calculus, ƒ (x) by r′(x); so that in equation (4), S is equal to the sum of infinitesimal elements of which F'(x) dx is the type; and therefore (7) becomes

[* x′(x) dx

= F(x) F(x).

(8)

5.] If the superior limit is x, x being a general value of the variable, subject to the condition that F'(x) is finite and continuous for all values of a between x

ΤΟ

and x,

then

F(x) dx = F(x) − F(X);

(9)

and omitting F(x), which is constant, the indefinite Integral of F'(x)dx is F(x); and we have

n

fx(x) dx

F′(x)dx = F(x).

(10)

Hence it follows that the definite integral of F'(x) dx between the limits x and x is the value of the indefinite integral when x = x,„, less its value when x = x; on this account it is frequently and conveniently expressed as follows

6.] Perhaps it may be supposed that the value of the definite integral depends on the number and magnitude of the elements

x-x-1, or on the mode of partition of

xx into its parts; if the elements however are infinitesimal, and their number consequently infinite, whatever is the mode of partition, the value of the indefinite integral is the same, as may thus be shewn :

Whatever another mode is, we may consider it to be a subdivision of the first, and thus its elements to be parts of the former elements. Suppose then a1- to be divided into n parts, and $1, 2, 3, E to be the values of a corresponding to the points of partition, and F(x) da to be the infinitesimal element: then the sum of all the infinitesimal elements corresponding to the successive values of a between x and x, is

...

+ F′(§n_1) (X1— §n−1) ;

F′(xo) (§1−xo)+F' (§1) (§2 − §1) + the value of which is, by (4) and (7), F(x1)-F(x). And as analogous results are true for each of the other elements x2-x1, ...x-x-1, so will the sum be true; and therefore equation (11) is true, independently of the particular mode of partition by which the elements are formed.

Hence we have finally, subject to the condition that F'(x) is finite and continuous within the limits x, and a。,

[ **'x' (x)dx =

F′(x ̧) (X1−x ̧q) + F′ (X1) (X2 −X1) +.....

... + F′(xn− 1 ) ( x n −xn−1) (12)

= F(x) F(x) + F(X2) — F(X1) + ... + F(x) — F (Xn_1)

= F(x) F(xg);

(13)

the sum given in the right-hand member of (12) being expressed by either the left-hand member of (12) or by the right-hand member of (13). It will be observed that in the series (12), the terms do not go as far as F'(x); in the definite integral therefore expressed by (13) the value of the element-function at the inferior limit is included, and that at the superior limit is excluded.

7.] To return to the consideration of the indefinite integral:

by equation (10) [ r'(x) dx = ¥ (x) ;

that is, the operation symbolized by dr, performed on r′(x), d changes it into F(r); but by the Differential Calculus is the dx

symbol of an operation which being performed on F (r) changes it into r'(x); therefore dæ and

d dx

are so related that one represents

a process the reverse of that represented by the other; that is,

d

according to the index law which the symbol is subject to,

-1

dx

Jdx = (d)1

= d-1dr:

dx

(14)

(15)

and represents an operation which is the reverse of differen

unity being used as a symbol of an operation which operating on a function leaves it unaltered.

Hence according to the notation of derived functions,

and in this mode of viewing the subject, the symbol de must be considered as a complex character, indicative of a certain analytical process to be performed on a certain function; the analytical process being the reverse of Derivation.

Hence the problem of Integration resolves itself into this; viz. to determine the function which, when differentiated, produces the infinitesimal element expressing the general term of the series; and therefore as this is a process the reverse of Differentiation, we may make use of our knowledge of the Differential Calculus, and as far as possible invert its rules; for these will thereby become those of the Integral Calculus; such processes we shall enter on in the next Chapter, and thereby obtain indefinite integrals, from which definite integrals may be deduced by means of equation (11).

In this aspect of the Calculus another point requires explana* See Vol. I. Art. 420.

tion. Since an arbitrary constant connected with a function of x by addition or subtraction disappears in Differentiation, so in the reverse process such a constant may be introduced; and thereby we have

fr'(x)dx

'(x)dx = F(x)+C.

The same result also follows from equation (9), wherein —F (x) being independent of x is constant with respect to it. But as the Integral Calculus might exist previously to the Differential, for the infinitesimal element may exist previously to and independently of the finite function, so its principles ought to have an independent basis. We shall therefore in the first place investigate certain properties of definite integrals, which will be required in the sequel, and also integrate from first principles some infinitesimal elements.

8.] THEOREM I.—If an infinitesimal element has a constant quantity as a factor, the definite integral will also have the same constant factor.

Let the infinitesimal element be a r'(x) dx, wherein a is a constant quantity; then

A constant factor therefore may be taken outside the sign of integration; and similarly may, if required, be removed from without to within the sign of Integration.

The following are particular cases:

The same theorem is of course true of an indefinite integral.

THEOREM II. The integral of the algebraic sum of any number of infinitesimal elements is equal to the algebraic sum of the integrals of the same elements.

Let F(x) dx, f'(x) dx, p'(x) dx, ... be any infinitesimal elements finite and continuous between the limits x, and xo; then