Volume of revolution

Calculate the volume of a solid of revolution using the disk/washer method or shell method.

Method f(x) = From x = to x =

Disk method (revolution around x-axis)

V = π ∫ₐᵇ [f(x)]² dx

Each cross-section is a circular disk with radius f(x).

Shell method (revolution around y-axis)

V = 2π ∫ₐᵇ x·f(x) dx

Each piece is a cylindrical shell with radius x and height f(x).

When to use which

Disk method is usually simpler when rotating around the x-axis. Shell method works better when rotating around the y-axis or when the function isn't easily inverted.