Fundamental Theorem of Calculus

The bridge connecting differentiation and integration — why antiderivatives compute areas and why the two operations are inverses of each other.

FTC Part 1: F(t) = ∫₀ᵗ f(x) dx — move t to see the accumulated area

f(x) = x² F(x) = x³/3

t = 2.504

The green curve F(t) tracks exactly the area under f — F′(t) = f(t)

Definition

The Fundamental Theorem of Calculus is two related results that together show differentiation and integration are inverse operations.

Part 1: If $f$ is continuous on $[a, b]$ and $F(x) = \int_a^x f(t)\,dt$ , then $F$ is differentiable and $F'(x) = f(x)$ .

In words: differentiation undoes integration. The area-accumulation function has derivative equal to the integrand.

Part 2: If $F$ is any antiderivative of $f$ (meaning $F' = f$ ), then:

$\int_a^b f(x)\,dx = F(b) - F(a)$

In words: to evaluate a definite integral, find any antiderivative and evaluate it at the endpoints.

Using FTC Part 2

$\displaystyle\int_0^3 x^2\,dx$

An antiderivative of $x^2$ is $F(x) = x^3/3$ .

$F(3) - F(0) = 27/3 - 0 = 9$

Try it

Evaluate $\displaystyle\int_1^e \frac{1}{x}\,dx$ .

Solution

Antiderivative: $\ln x$ .

$[\ln x]_1^e = \ln e - \ln 1 = 1 - 0 = 1$

Related concepts