AP Calculus AB / Integration and Accumulation of Change
Lesson
The Fundamental Theorem and undoing a derivative
An antiderivative turns a definite integral into F(b) − F(a). The other half differentiates an integral with a variable limit.
Learning goals
- Evaluate a definite integral with an antiderivative.
- Differentiate a function defined as an integral with a variable upper limit.
Explanation
If F′ = f and f is continuous on [a, b], then ∫_a^b f(x) dx = F(b) − F(a). That is the evaluation half of the Fundamental Theorem. Substitution is the chain rule run backward: if the inside is messy, let u be the inside and remember du.
The other half says that d/dx of ∫_a^x f(t) dt equals f(x). If the upper limit is a function, chain-rule it. The dummy variable t is not the same as the free variable x. Mixing them is a notation injury, not a style choice.
Key terms
- Antiderivative. A function F whose derivative is the given f.
- Fundamental Theorem of Calculus. The pair of results that connect derivatives and definite integrals: evaluation by antiderivatives, and differentiation of an integral with a variable limit.
Common mistakes
- Forgetting to multiply by the derivative of a variable limit.
- Changing u but leaving the old x-limits on a definite integral.
Practice
Original Marlow Works items. Check the answer explanation after you try.
Water enters a tank at a rate r(t) = 8 − 0.5t liters per minute for 0 ≤ t ≤ 10. At t = 0 the tank holds 20 liters. (a) Write an expression for the volume V(t). (b) Find V(6). (c) Interpret V(6) − V(0) in context.
Original Marlow Works item — not a College Board question.
Take your time—this is practice, not a test.
Answer explanation
If F(x) = ∫_0^x (3t² + 1) dt, what is F′(2)?
Answer. 3(2)² + 1 = 13.
F′(x) = 3x² + 1, so plug in 2. You do not need F(2).
Related resources
External links with reuse status. Marlow Works is independent and does not copy restricted exam or textbook material.
Official / link only
AP Calculus AB course page
Official eight-unit framework and 2027 exam updates.
College Board · All rights reserved · accessed 2026-10-01
Open sourceReference only — do not copy
OpenStax College Algebra 2e
Algebra reference; current terms include noncommercial and AI-ingestion limits.
OpenStax · Restricted — see publisher page · accessed 2026-10-01
Open source