AP Calculus AB Unit 8 Practice Test: Applications of Integration

Putting Integration to Work

Unit 8 is the final and most applied unit in AP Calculus AB. It draws on every integration skill from Unit 6 and combines them with geometric and physical reasoning to solve problems involving areas, volumes, average values, and motion. FRQs in this unit are typically multi-part and require careful setup before any computation begins.

Topics in This Unit Practice Test

• Average value of a function over an interval: (1/(b−a)) ∫[a to b] f(x) dx
• Area between two curves using definite integrals
• Setting up area problems when curves intersect at multiple points
• Volumes of solids of revolution using the disc method
• Volumes using the washer method when the region has a hole
• Volumes of solids with known cross-sections (squares, semicircles, triangles)
• Motion problems: displacement vs. total distance traveled
• Accumulation problems: interpreting ∫[a to b] f′(t) dt as net change

Area Between Curves: AP FRQ Approach

Area problems require you to identify which function is on top, find the limits of integration (often by solving for intersection points), and integrate the difference of the two functions. When the curves switch positions within the interval, you must split the integral at each crossing point and ensure the integrand is always non-negative in each sub-interval. Setting up the integral correctly is worth more AP points than the final numerical answer.

Disc and Washer Methods for Volumes

When a region is rotated around an axis, the resulting solid's volume is computed using the disc or washer method. The disc method applies when the region touches the axis of rotation; the washer method applies when there is a gap between the region and the axis, creating a hole in the solid. The integrand for the washer method is π(R² − r²), where R is the outer radius and r is the inner radius as functions of the integration variable. A frequent AP error is squaring the difference of functions rather than taking the difference of their squares.

Volumes with Known Cross-Sections

These problems describe a solid whose cross-sections perpendicular to an axis have a known shape — typically a square, semicircle, equilateral triangle, or right isosceles triangle. The volume is the integral of the cross-sectional area as the cross-section sweeps along the axis. The key step is expressing the area of one cross-section in terms of the functions that define the base of the solid.

Displacement vs. Total Distance in Motion Problems

Displacement is ∫[a to b] v(t) dt — which accounts for direction and may involve cancellation. Total distance is ∫[a to b] |v(t)| dt — which requires splitting the integral at any time when v(t) = 0 and the particle reverses direction. Confusing these two quantities is one of the most common errors in AP Calculus AB motion FRQs.