Inside This Unit: The Full Breakdown
This unit applies definite integrals to compute areas between curves, volumes of solids of revolution using disc and washer methods, and volumes of solids with known cross sections.
Why it matters
Applications of integration are among the highest-scoring free-response topics. Volume problems using discs, washers, and cross sections appear almost every year. These problems test your ability to set up integrals correctly, which is the skill the AP exam values most.
Key concepts
- Area between curves equals the integral of (top function minus bottom function) over the interval of intersection.
- The disc method computes volume by rotating a region around an axis: V = pi * integral of [R(x)]^2 dx.
- The washer method handles regions with holes: V = pi * integral of ([R(x)]^2 - [r(x)]^2) dx.
- Cross-section volumes use V = integral of A(x) dx, where A(x) is the area of a cross-sectional slice.
Area Between Curves
To find the area between two curves, integrate the difference of the top and bottom functions over the interval where they intersect. First find the intersection points by setting the functions equal. If the curves switch which is on top, split into subintervals. For regions bounded on the left and right rather than top and bottom, integrate with respect to y instead. Always sketch the region to identify the correct integrand. The AP exam frequently presents these problems with functions defined by tables or graphs rather than explicit formulas.
Volumes of Revolution: Disc and Washer Methods
When a region is rotated around an axis, the resulting solid has circular cross sections. The disc method applies when the region touches the axis of rotation: V = pi * integral of [R(x)]^2 dx, where R(x) is the distance from the curve to the axis. The washer method applies when there is a gap between the region and the axis, creating a hollow center: V = pi * integral of ([R(x)]^2 - [r(x)]^2) dx. Choosing whether to integrate with respect to x or y depends on the axis of rotation and which setup avoids splitting the integral.
Volumes with Known Cross Sections
Some solids are not formed by rotation but have known cross-sectional shapes — squares, semicircles, equilateral triangles, or rectangles — perpendicular to an axis. The volume is V = integral of A(x) dx, where A(x) is the area of the cross section at position x. The base of the solid is typically a region in the xy-plane, and the side length of each cross section equals the distance between the bounding curves. These problems are highly formulaic once you identify the cross-sectional shape and express its area in terms of x.
AP exam tip
On volume free-response questions, always write the integral setup with correct limits, integrand, and dx or dy before evaluating. The setup is worth more points than the final numerical answer.
Connections to other units
- Unit 5: Setting up definite integrals relies on accumulation concepts and the Fundamental Theorem.
- Unit 4: Analyzing the geometry of the region connects to function analysis with derivatives.
- Unit 6: Some volume problems are set up using solutions to differential equations as bounding curves.
