What is the Fundamental Theorem of Calculus?

∫ₐᵇ f(x) dx = F(b) - F(a)

where F'(x) = f(x). The most useful result in calculus.

The Fundamental Theorem of Calculus connects two apparently separate ideas. Part 1: if you integrate a function from a fixed point to x, the derivative of that integral is the original function. Part 2: the definite integral of f from a to b equals any antiderivative F evaluated at b minus F at a.

Area under x² from 0 to 2: antiderivative gives exact answer

∫₀² x² dx = [x³/3]₀² = 8/3 − 0 = 8/3 ≈ 2.667. The antiderivative F(x) = x³/3 gives the exact area without approximation.

Before this theorem, computing areas required Riemann sums: dividing the region into thin rectangles, summing them all, and taking the limit. The FTC replaces all of that with one subtraction. Newton understood this by 1666 and Leibniz independently by 1675. Their dispute over priority split European and British mathematics for a generation.

The two parts of the Fundamental Theorem of Calculus

Part 1: d/dx [∫ₐˣ f(t)dt] = f(x)

Part 2: ∫ₐᵇ f(x)dx = F(b) − F(a) where F'(x) = f(x)

Part 1 says differentiation undoes integration. Part 2 says to evaluate an integral, find an antiderivative and subtract the endpoints.

Every integral taught in calculus courses uses Part 2: find an antiderivative, evaluate at the endpoints, subtract. This works because differentiation and integration are exact inverses of each other. It is one of the deepest and most useful results in all of mathematics.

Taylor Series → Wallis Product →

Riemann sum approximating area: FTC replaces this with one calculation

A Riemann sum with 8 rectangles gives ≈ 0.273. The exact answer is 8/3 ≈ 2.667. The Fundamental Theorem gives exact results with no rectangles needed.

Physics application

Work done by a variable force F(x) over displacement from a to b is W = integral from a to b of F(x) dx = P(b) - P(a), where P is the potential energy function satisfying P' = -F. Velocity integrates to displacement; force integrates to impulse. The FTC is what makes these calculations tractable rather than requiring infinite Riemann sums.