The germs of this method of formulating the problem of quadratures are found in the writings of Archimedes. The method leads to a definition of a definite integral, but the direct application of it to the evaluation of integrals is in general difficult. Any process for evaluating a definite integral is a process of integration, and the rules for evaluating integrals constitute the integral calculus.
| Fig. 4. |
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 Theorem of Inversion. integral ∫ξa ƒ(x)dx, and it is a function of ξ. Let BD be displaced to B′D′ so that ξ becomes ξ + δξ (see fig. 4). The area of the figure ACD′B′ is represented by the integral ∫ξ+Δξa ƒ(x)dx, and the increment ΔA of the area is given by the formula
ΔA = ∫ξ+Δξξ ƒ(x) dx,
which represents the area BDD′B′. This area is intermediate between those of two rectangles, having as a common base the segment BB′, and as heights the greatest and least ordinates of points on the arc DD′ of the curve. Let these heights be H and h. Then ΔA is intermediate between HΔξ and hΔξ, and the quotient of differences ΔA/Δξ is intermediate between H and h. If the function ƒ(x) is continuous at B (see Function), then, as Δξ is diminished without limit, H and h tend to BD, or ƒ(ξ), as a limit, and we have
| dA | = ƒ(ξ). |
| dξ |
The introduction of the process of differentiation, together with the theorem here proved, placed the solution of the problem of quadratures on a new basis. It appears that we can always find the area A if we know a function F(x) which has ƒ(x) as its differential coefficient. If ƒ(x) is continuous between a and b, we can prove that
A = ∫ba ƒ(x) dx = F(b) − F(a).
When we recognize a function F(x) which has the property expressed by the equation
| dF(x) | = ƒ(x), |
| dx |