The Fundamental Theorem of Calculus

The Bridge Between the Two Ideas

So far integration has worn two faces — antiderivatives (reversing differentiation) and definite integrals (accumulated area). The Fundamental Theorem of Calculus (FTC) reveals that these are two sides of one coin. It is arguably the most important theorem in calculus, uniting the subject's two halves.

The theorem has two parts:

• Part 1 — if you define an area-accumulating function $F(x) = \int_a^x f(t)\, dt$, then its derivative is the original function: $F'(x) = f(x)$. 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. It means you don't 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 at all.

Conceptually, the FTC says accumulation and rate-of-change are inverse operations. This is why velocity integrates to displacement, why a rate integrates to a total, and why the whole edifice of calculus hangs together. The two operations you learned separately are revealed as one.

Common pitfall: in Part 2, subtracting the endpoints in the wrong order, or forgetting it applies to the definite integral only. The rule is $F(b) - F(a)$ — the upper limit minus the lower limit; reversing them flips the sign. And the $+C$ of the antiderivative does not matter here (it cancels in the subtraction), which is why any antiderivative works.