AP Calculus AB and BC glossary

Shell method

Also called: Cylindrical shells

The shell method computes a volume of revolution by slicing the region parallel to the axis of rotation and treating each slice as a thin cylindrical shell. Its integrand is two pi times the radius times the height.

V = 2\pi \int_a^b r(x)\,h(x)\,dx

The factor of $2\pi r$ is the circumference of the shell and $h$ is its height, so the product is the area of the unrolled rectangle. Multiplying by the thickness and integrating gives the volume.

Shells are the better choice when slicing perpendicular to the axis would force you to solve the equation for the other variable, which is often messy or impossible.

Appears in: Unit 8: Applications of Integration