AP Calculus AB and BC

Washer vs Shell Method

Washers slice perpendicular to the axis of rotation and integrate in the axis variable. Shells slice parallel to it and integrate in the other variable. Choose whichever lets you avoid rewriting the equation in terms of a variable it is not already solved for.

Washer method

Use when: The curves are already expressed in the variable you would be integrating in for a perpendicular slice.

Shell method

Use when: A perpendicular slice would force you to solve the equation for the other variable.

Side by side

	Washer	Shell
Slice direction	Perpendicular to the axis	Parallel to the axis
Integrand	$\pi\left(R^2 - r^2\right)$	$2\pi r h$
Rotating about the $y$ axis	Integrate in $y$	Integrate in $x$
Needs solving for the other variable	Often	Often not

Both give the same volume, so the choice is purely about which integral you can actually evaluate. Sketch a representative slice in each direction and see which one keeps the functions in the form you already have.

The shell integrand is the circumference of the unrolled shell, $2\pi r$ , times its height. Thinking of it as a flattened rectangle explains why there is no squaring and no subtraction.

Radius is still a distance

In both methods the radius is measured from the axis of rotation, so a shift in the axis changes the expression even though the region is unchanged.

Frequently asked questions

Is the shell method on the AP exam?

Shells are not required by the CED, but the method is often the fastest route and is fully valid on free response.

In the CED: Unit 8: Applications of Integration