Short answer
The Fundamental Theorem of Calculus links area accumulation to antiderivatives: if F' = f, then integral from a to b of f(x) dx equals F(b) - F(a).
- Definite integrals represent signed accumulation.
- FTC turns many definite integrals into endpoint evaluation.
- Keep bounds and antiderivative parentheses clear.
Understand accumulation and sign
A definite integral accumulates net change. Area above the axis contributes positively, and area below contributes negatively.
This signed interpretation explains why a large geometric area can still yield a small net integral.
Use antiderivatives with bounds
Find an antiderivative F(x), then compute F(b)-F(a). Keep exact values when possible before decimal approximation.
Indefinite integrals need +C, but definite integrals do not keep +C because constants cancel in subtraction.
- Reverse bounds changes the sign of the integral.
- Integral over zero-width interval is zero.
- Linearity allows splitting complex sums into simpler terms.
Evaluate a polynomial definite integral
Compute integral from 1 to 3 of (2x + 1) dx.
- Find antiderivative: integral of (2x+1) is x^2 + x.
- Evaluate upper bound: F(3) = 9 + 3 = 12.
- Evaluate lower bound and subtract: F(1) = 1 + 1 = 2, so 12 - 2.
Result: The integral value is 10.
Common mistakes
- Forgetting to subtract lower-bound value.
- Adding +C to a definite integral result.
- Dropping parentheses around F(b)-F(a).
- Confusing geometric area with signed integral value.
Try one
If F' = f, how do you compute integral from a to b of f(x) dx?
Compute F(b) - F(a).
Sources
- OpenStax Calculus Volume 1Open textbook covering limits, derivatives, applications, and introductory integration.
- College Board AP CalculusOfficial AP Calculus AB course overview and unit structure.