Britannica Sources used in this answer

How does the fundamental theorem of calculus make integration easier?

Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.

transformation of a circular region into an approximately rectangular region — analysis \
This suggests that the same constant (π) appears in the formula for the circumference, 2πr, and in the formula for the area, πr². As the number of pieces increases (from left to right), the “rectangle” converges on a πr by r rectangle with area πr²—the same area as that of the circle. This method of approximating a (complex) region by dividing it into simpler regions dates from antiquity and reappears in the calculus.

The fundamental theorem of calculus makes integration significantly easier by establishing an inverse relationship between differentiation and integration.1

Rather than relying on complex summation methods to find the area under a curve, the theorem allows one to solve definite integrals through antidifferentiation. Specifically, if you can identify an antiderivative $F$ for a continuous function $f$ (where the derivative $F^{'}$ equals $f$ ), the definite integral over an interval $[a, b]$ is simply the difference $F (b) - F (a)$ .21

This approach transforms the problem of calculating an integral into the task of finding a function whose derivative is the given function. By providing this systematic method, the theorem serves as the backbone of calculus, enabling mathematicians and scientists to evaluate areas, volumes, and other quantities efficiently by leveraging known derivative rules.231