The Fundamental Theorem of Calculus

Uniting Calculus

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.

Conceptually, accumulation and rate-of-change are inverse operations. This is why velocity integrates to displacement and any rate integrates to a total.

The Evaluation Rule

Part 2 is the practical powerhouse. You never need Riemann sums for exact integrals; just find an antiderivative and subtract endpoint values.

Worked Example: Evaluate $\int_0^2 x\, dx$.

1. Find antiderivative: $F(x) = \frac{x^2}{2}$.
2. Subtract limits: $\left[\frac{x^2}{2}\right]_0^2 = \frac{2^2}{2} - \frac{0^2}{2} = 2 - 0 = 2$.

Common Pitfall: Subtracting endpoints in the wrong order. The rule is $F(b) - F(a)$ (upper minus lower); reversing it flips the sign.

Note that any antiderivative works. The $+C$ always cancels out during subtraction $F(b) - F(a)$.