“ Any process for evaluating a definite integral is a process of integration, and the rules for evaluating integrals constitute the integral calculus.
6. The chief of these rules is obtained by regarding the extreme ordinate BD as variable. Let ξ now denote the abscissa of B. The area A of the figure ACDB is represented by the integral , and it is a function of ξ. Let BD be displaced to B′D′ so that becomes (seeTheorem of Inversion. fig. 4) . The area of the figure ACD′B′ is represented by the integral and the increment ΔA is given by the formula:
which represents the area BDD′B′. ”

“ The fundamental artifice of the calculus is the artifice of forming differentials without first forming differential coefficients. From an equation containing x and y we can deduce a new equation, containing also Δx and Δy, by substituting x+Δx for x and y+Δy for y. If there is a differential coefficient Fundamental Artifice.of y with respect to x, then Δy can be expressed in the form φ.Δx + R, where lim.Δx = 0 (R/Δx) = 0, as in § 7 above. ”

“ For the problem of differentiating such functions reference may be made to Function.
12. The processes of the integral calculus consist largely in transformations of the functions to be integrated into such forms that they can be recognized as differential coefficients of functions which have previously been differentiated. Corresponding to the results in the table of § 11 we Indefinite Integrals. ”

“ The limit expressed by the right-hand member of this defining equation is often written
and is called the “derived function” of ƒ (x) , sometimes the “derivative” or “derivate” of ƒ (x) . When the function ƒ (x) is a rational integral function, the division by Δx can be performed, and the limit is found by substituting zero for Δx in the quotient. For example, if ƒ (x) = x2, we have
and
The process of forming the derived function of a given function is called differentiation. ”