AP Calculus BC Unit 6: Integration & Accumulation
Study Riemann sums, FTC, u-substitution, integration by parts with exam-format practice and rubric-based scoring.
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Inside This Unit: The Full Breakdown
Integration computes accumulated change. This unit covers antiderivatives, Riemann sums, the Fundamental Theorem of Calculus, and u-substitution as the primary integration technique.
Why it matters
The FTC and u-substitution are foundational for all remaining BC topics. Series representations of functions, parametric arc length, and polar area all build on definite integration. Mastery here is non-negotiable.
Key concepts
- Antiderivatives reverse differentiation: if F' = f, then integral of f dx = F + C.
- Riemann sums (left, right, midpoint, trapezoidal) approximate definite integrals.
- FTC Part 1: d/dx[integral from a to x of f(t) dt] = f(x).
- FTC Part 2: integral from a to b of f(x) dx = F(b) - F(a).
Antiderivatives and Basic Integration
Every derivative rule has a corresponding antiderivative: integral of x^n dx = x^(n+1)/(n+1) + C for n != -1, integral of 1/x dx = ln|x| + C, integral of e^x dx = e^x + C, and so on. The constant of integration C accounts for the family of functions sharing the same derivative. Initial conditions pin down a specific value of C. In BC, you also need antiderivatives of inverse trig forms: integral of 1/(1+x^2) dx = arctan(x) + C.
The Fundamental Theorem of Calculus
FTC Part 1 says differentiation undoes integration: the derivative of the integral from a to x of f(t) dt is f(x). When the upper limit is a function g(x), apply the chain rule to get f(g(x)) * g'(x). FTC Part 2 provides the evaluation shortcut: integrate using an antiderivative and subtract endpoint values. Together these theorems make computing definite integrals practical and connect the two branches of calculus.
U-Substitution
U-substitution reverses the chain rule. Set u = g(x) so that du = g'(x) dx, transforming a complicated integral into a simpler one in u. For definite integrals, convert the limits to u-values so you never need to substitute back. Common substitutions include u = inner function of a composition. In BC, u-substitution is the first technique to try before considering integration by parts or partial fractions.
AP exam tip
On definite integral problems, changing the limits of integration when you substitute eliminates the need to back-substitute — this saves time and reduces errors on the exam.
Connections to other units
- Unit 1: The definite integral is defined as a limit of Riemann sums.
- Unit 7: Differential equations require antidifferentiation to solve.
- Unit 10: Power series are integrated term by term using basic antiderivative rules.
