Washer Method

College/University

Definition

If you wanted to find the volume of a round shape with a hole in the center, the washer method can be used. This technique makes use of the disk method. A shape is cut into thin pieces (or disks) and then you can find the volume of the slices by subtracting out the hole in the middle.

Worked examples

\(V = \pi \int_{a}^{b} \left[ R(x)^2 - r(x)^2 \right] dx\)

Volume equals π times the integral of outer radius squared minus inner radius squared.

\(V = \pi \int_{0}^{1} \left[ (2)^2 - (x^2)^2 \right] dx = \pi \int_{0}^{1} (4 - x^4) dx\)

Revolving the region between \(y=x^2\) and \(y=2\) around the x-axis creates washers with outer radius 2 and inner radius \(x^2\).

Common mistakes

\(V = \pi \int_{a}^{b} [R(x) - r(x)]^2 dx\) → \(V = \pi \int_{a}^{b} \left[ R(x)^2 - r(x)^2 \right] dx\) Square each radius separately before subtracting; don't subtract first then square.
\(V = \pi \int_{a}^{b} \left[ r(x)^2 - R(x)^2 \right] dx\) → \(V = \pi \int_{a}^{b} \left[ R(x)^2 - r(x)^2 \right] dx\) Always subtract inner radius squared from outer radius squared to keep volume positive.
Using washer method when there is no hole in the solid → Use disk method when the region touches the axis of rotation Washer method is only for solids with a hollow center; otherwise the inner radius is zero.

Where you'll use it next

You'll apply the washer method throughout Calculus II when finding volumes of revolution for regions that don't touch the axis, and it extends to shell method problems and real engineering applications modeling hollow objects.

Found in 1 StudyPug lesson

Mastering the Disc Method for Volumes of Revolution

Calculus 2

Unlock the power of the disc method to calculate volumes of solids of revolution. Perfect your calculus skills with our comprehensive guide, from basic concepts to advanced applications.

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