AP Calculus BC Unit 8: Applications of Integration
Study area, volume, arc length, improper integrals with exam-format practice and rubric-based scoring.
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Inside This Unit: The Full Breakdown
Definite integrals compute areas between curves, volumes of solids of revolution, and volumes with known cross sections. BC adds integration by parts and partial fractions as additional techniques.
Why it matters
Volume and area problems appear on nearly every AP exam. BC students must also master integration by parts and partial fractions, which are essential for evaluating integrals that arise in series, arc length, and improper integral problems.
Key concepts
- Area between curves = integral of (top - bottom) or (right - left).
- Disc/washer methods compute volumes of revolution: V = pi * integral of R^2 or (R^2 - r^2).
- Integration by parts: integral of u dv = uv - integral of v du (BC only).
- Partial fractions decompose rational expressions into simpler fractions for integration (BC only).
Areas and Volumes
Area between curves uses the integral of (top - bottom) dx or (right - left) dy over the appropriate interval. For volumes, the disc method rotates a region touching the axis: V = pi * integral of [R(x)]^2 dx. The washer method handles gaps: V = pi * integral of ([R]^2 - [r]^2) dx. Cross-section volumes use V = integral of A(x) dx for known shapes like squares, semicircles, or triangles. These setups are heavily tested on the free response.
Integration by Parts
Integration by parts reverses the product rule: integral of u dv = uv - integral of v du. Choose u using LIATE priority (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential). Common applications include integrals of x*e^x, x*sin(x), and ln(x). Sometimes you apply it twice and solve for the original integral algebraically. This technique is essential in BC for evaluating certain improper integrals and integrals arising in series convergence problems.
Partial Fractions
Partial fraction decomposition breaks a rational function P(x)/Q(x) into a sum of simpler fractions, each of which is easy to integrate. For distinct linear factors, write A/(x-a) + B/(x-b) and solve for A and B. For repeated linear factors, include terms with increasing powers in the denominator. The technique applies only when the degree of the numerator is less than the degree of the denominator — otherwise, perform polynomial long division first. This is a BC-only technique tested on both multiple choice and free response.
AP exam tip
For integration by parts, write out your choice of u and dv clearly before computing. If your first choice leads to a more complicated integral, switch your assignment and try again.
Connections to other units
- Unit 5: All volume and area setups rely on the definite integral and the FTC.
- Unit 9: Arc length of parametric curves requires integration techniques from this unit.
- Unit 10: Integration by parts is used to derive certain series and evaluate remainders.
