AP Calculus AB Unit 6: Integration & Accumulation
Study Riemann sums, definite integrals, FTC, antiderivatives with exam-format practice and rubric-based scoring.
Start AP Calc AB Practice · Full Study Guide
AP and Advanced Placement are trademarks of College Board. AimFive is not affiliated with or endorsed by College Board.
Inside This Unit: The Full Breakdown
Integration reverses differentiation. This unit introduces antiderivatives, definite integrals as accumulated change, Riemann sums, and the Fundamental Theorem of Calculus that connects derivatives and integrals.
Why it matters
The Fundamental Theorem of Calculus is the most important result in the course. It bridges the two halves of calculus and appears in nearly every free-response question involving integrals. Riemann sum approximations and accumulation interpretations are also heavily tested.
Key concepts
- An antiderivative of f is a function F such that F'(x) = f(x). The general antiderivative includes + C.
- A Riemann sum approximates a definite integral using rectangles (left, right, midpoint) or trapezoids.
- The Fundamental Theorem of Calculus Part 1: d/dx[integral from a to x of f(t) dt] = f(x).
- The Fundamental Theorem Part 2: integral from a to b of f(x) dx = F(b) - F(a) where F' = f.
Antiderivatives and Indefinite Integrals
Finding an antiderivative means reversing differentiation. If F'(x) = f(x), then F is an antiderivative of f. Since the derivative of a constant is zero, antiderivatives are only determined up to a constant: the integral of f(x) dx = F(x) + C. Basic antiderivative rules mirror derivative rules in reverse: the integral of x^n is x^(n+1)/(n+1) for n not equal to -1, the integral of 1/x is ln|x|, the integral of e^x is e^x, and so on. U-substitution extends these rules to composite functions.
Riemann Sums and Definite Integrals
A definite integral computes the net signed area between a curve and the x-axis. Riemann sums approximate this area using rectangles. Left Riemann sums use the left endpoint of each subinterval for the height, right sums use the right endpoint, and midpoint sums use the center. Trapezoidal sums average the left and right values. As the number of subintervals increases, these approximations converge to the exact integral. The AP exam tests whether each approximation overestimates or underestimates based on whether the function is increasing or decreasing, concave up or concave down.
The Fundamental Theorem of Calculus
The FTC connects differentiation and integration. Part 1 says that if g(x) = integral from a to x of f(t) dt, then g'(x) = f(x) — the derivative of an accumulation function returns the original integrand. Part 2 says that the definite integral from a to b of f(x) dx equals F(b) - F(a), where F is any antiderivative of f. Together these results mean that integration and differentiation are inverse operations. The FTC Part 1 combined with the chain rule appears frequently on AP free-response questions involving accumulation functions.
AP exam tip
When applying FTC Part 1 with a variable upper limit like g(x) = integral from 0 to x^2 of f(t) dt, remember to apply the chain rule: g'(x) = f(x^2) * 2x.
Connections to other units
- Unit 1: The definite integral is defined as a limit of Riemann sums, tying integration back to limits.
- Unit 4: Analyzing accumulation functions uses first and second derivative analysis from the analytical applications unit.
- Unit 8: Applications of integration extend the definite integral to compute areas, volumes, and accumulated quantities.
