AP Calculus AB and BC

Disk vs Washer Method

Use the disk method when the region being rotated touches the axis of rotation, so each slice is a solid circle. Use the washer method when a gap separates the region from the axis, because that gap becomes a hole and each slice becomes a ring.

Disk method

Use when: The region is bounded by the axis of rotation itself, so no gap exists.

Washer method

Use when: The region is bounded away from the axis, or lies between two curves.

Side by side

	Disk	Washer
Slice shape	Solid circle	Ring
Integrand	$\pi R^2$	$\pi\left(R^2 - r^2\right)$
Region touches the axis	Yes	No
Radii needed	One	Two

A disk is just a washer with inner radius zero, so there is really one method with a special case. Sketching the region and asking whether any gap separates it from the axis settles the choice in seconds.

Both radii are distances from the axis, so rotating about a line other than an axis changes every expression. Rotating about $y = 4$ makes the outer radius $4 - f(x)$ rather than $f(x)$ .

The mistake

Writing $(R - r)^2$ instead of $R^2 - r^2$ . Those are different quantities, and only the second is the area of a ring.

Frequently asked questions

How do I know which radius is outer?

The outer radius reaches the curve farther from the axis of rotation. Sketch a representative slice and measure both distances from the axis.

In the CED: Unit 8: Applications of Integration