Volumes — Disk & Washer Methods

Disk method (no hole):

V = π ∫ from a to b [f(x)]² dx

Rotate f(x) around the x-axis. Cross-sections are circles.

Washer method (with hole):

V = π ∫ from a to b ([R(x)]² - [r(x)]²) dx

R = outer radius, r = inner radius.

Revolving around y-axis: Integrate with respect to y, or use the shell method.

Shell method:

V = 2π ∫ from a to b x · f(x) dx

Useful when rotating around the y-axis but the function is in terms of x.

AP exam: Volumes appear on free response every year. Set up the integral correctly and show your work. Partial credit is available for correct setup even with computation errors.

Reference:

Wikipedia: Volumes of Revolution

https://en.wikipedia.org/wiki/Solid_of_revolution