The Fundamental Theorem of Calculus

The Bridge Between the Two Ideas

Integration has worn two faces — antiderivatives (reversing differentiation) and definite integrals (accumulated area). The Fundamental Theorem of Calculus (FTC) reveals they are two sides of one coin. It is the central theorem of calculus, uniting its two halves.

The theorem has two parts:

• Part 1 — if $F(x) = \int_a^x f(t)\, dt$ (an area-accumulating function), then $F'(x) = f(x)$. In words, the derivative of the accumulated area is the height of the curve: differentiation and integration undo each other.
• Part 2 (the evaluation rule) — to compute a definite integral, find any antiderivative $F$ and take the difference at the endpoints:

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

Part 2 is the practical powerhouse. You never need Riemann sums to evaluate an integral exactly — just find an antiderivative and subtract its endpoint values. For example $\int_0^2 x\, dx = \left[\frac{x^2}{2}\right]_0^2 = 2 - 0 = 2$, matching the triangle's area, with no summation.

Conceptually, the FTC says accumulation and rate-of-change are inverse operations. This is why velocity integrates to displacement, why any rate integrates to a total, and why the two operations learned separately turn out to be one. The $+C$ of the antiderivative does not matter in Part 2 — it cancels in the subtraction $F(b) - F(a)$ — which is exactly why any antiderivative works.

Common pitfall: in Part 2, subtracting the endpoints in the wrong order. The rule is $F(b) - F(a)$ — the upper limit minus the lower limit; reversing them flips the sign of the answer. And remember it evaluates the definite integral: any antiderivative works (the $+C$ cancels), so do not fret over which one to choose.