Calculus BC · Integration and Accumulation of Change · 14 min read · Updated 2026-05-11
FTC and Definite Integrals — AP Calculus BC
AP Calculus BC · Integration and Accumulation of Change · 14 min read
1. FTC Part 1: Derivatives of Accumulation Functions★★☆☆☆⏱ 4 min
The first part of FTC connects differentiation and accumulation of area under a curve. It establishes that the derivative of an accumulation function (a function defined by an integral with a variable bound) is exactly the original integrand.
Exam tip: If both bounds are variable, split the integral at any constant between the bounds, then apply FTC 1 to each piece. Do not rely on memorization alone for this case.
2. FTC Part 2: Evaluating Definite Integrals★★☆☆☆⏱ 4 min
The second part of FTC lets us compute definite integrals using antiderivatives, eliminating the need for tedious Riemann sum limit calculations. It is the primary method for evaluating definite integrals in AP Calculus.
Exam tip: Always evaluate the upper bound first, then subtract the lower bound. Reversing the order is the most common source of sign errors on definite integral problems.
3. The Net Change Theorem★★★☆☆⏱ 3 min
The Net Change Theorem is a direct applied interpretation of FTC Part 2, and it is extremely common in AP free-response questions involving rates of change.
Exam tip: Always include units in your final answer for contextual net change problems. AP graders routinely deduct 1 point for missing or incorrect units.
4. Additional AP-Style Worked Examples★★★☆☆⏱ 3 min
Common Pitfalls
Why: Confuses variable upper limits with variable lower limits, forgetting reversing integration order changes the integral's sign.
Why: Rushes to plug in values without reading the bounds, confuses the statement of FTC 2.
Why: Confuses the dummy variable of integration with the variable bound, treats the variable of integration as a constant.
Why: Confuses definite integrals (which evaluate to a single number) with indefinite integrals (which are families of antiderivatives).
Why: Forgets that FTC only applies if the integrand is continuous on the entire interval between the two bounds.
Why: Confuses net change (integral of the rate) with total change (integral of the absolute value of the rate).