Unit 6 Practice Test: Integration and Accumulation of Change (AP Calculus BC)

Integration in AP Calculus BC

Unit 6 is the integration foundation of AP Calculus BC and covers significantly more ground than the corresponding unit in AP Calculus AB. BC students are expected to master not only the techniques covered in AB — Riemann sums, the Fundamental Theorem of Calculus, and u-substitution — but also advanced integration methods including integration by parts and partial fraction decomposition. These BC-specific techniques appear directly in FRQ questions and in the integration problems embedded in Units 7–10.

Core Topics in Unit 6

Riemann Sums and Approximating Area

Left, right, midpoint, and trapezoidal Riemann sums approximate the definite integral. AP questions ask students to calculate these approximations from function values or tables and to determine whether an approximation is an overestimate or underestimate based on function behavior (increasing/decreasing, concave up/down).

The Fundamental Theorem of Calculus

Both parts of the FTC are tested. Part 1 establishes that differentiation and integration are inverse operations and is used to differentiate functions defined by integrals with variable upper limits. Part 2 provides the method for evaluating definite integrals using antiderivatives. FTC Part 1 with chain rule applied to the upper limit is a frequent AP multiple-choice topic.

Antiderivatives and Indefinite Integrals

Students must know antiderivative rules for polynomial, exponential, logarithmic, and trigonometric functions. Recognizing the appropriate antiderivative form quickly is essential in timed exam conditions.

U-Substitution

U-substitution is the primary technique for integrating composite functions. It requires identifying the inner function, computing du, rewriting the integrand entirely in terms of u, integrating, and back-substituting. Both definite and indefinite versions are tested. Changing limits of integration for definite integrals using u-substitution is a common error point.

Integration by Parts (BC)

Integration by parts uses the formula ∫u dv = uv − ∫v du and applies when the integrand is a product of two functions from different categories (polynomial × exponential, polynomial × logarithm, etc.). Choosing u and dv correctly is the critical decision. Some problems require applying integration by parts more than once. This technique is BC-exclusive and appears in both MCQ and FRQ sections.

Partial Fraction Decomposition (BC)

Partial fractions decompose a rational integrand into simpler fractions that can each be integrated separately. This technique is particularly important in Unit 7 for solving separable differential equations and in Unit 10 for certain series representations. Setting up the decomposition and solving for coefficients are both AP-tested skills.

Why BC Integration Techniques Matter Beyond Unit 6

Integration by parts appears in the solution of differential equations, in arc length calculations in Unit 8, and in series derivations in Unit 10. Partial fractions appear in logistic differential equation solutions in Unit 7. Fluency with all Unit 6 techniques is not optional for BC students — it directly enables performance in every subsequent unit.

Unit 6 Practice Test Coverage

Practice questions span Riemann sum approximation and overestimate/underestimate analysis, FTC Part 1 and Part 2 applications including chain rule variants, u-substitution for definite and indefinite integrals, integration by parts with single and repeated application, and partial fraction decomposition with integration. Both MCQ and FRQ question formats are represented.