AP Calculus AB and BC glossary

Washer method

The washer method computes the volume of a solid of revolution with a hole through it by subtracting the inner circle's area from the outer circle's area on every slice. It is used whenever the rotated region does not touch the axis of rotation.

V=π∫ab[(R(x))2−(r(x))2]dxV = \pi \int_a^b \left[\left(R(x)\right)^2 - \left(r(x)\right)^2\right] dx

The outer radius reaches the curve farther from the axis and the inner radius reaches the nearer one. Both are distances from the axis, so rotating about a line other than an axis changes both expressions.

The mistake

Writing (R−r)2(R - r)^2 instead of R2−r2R^2 - r^2. Those are different quantities, and only the second describes the area of an annulus.

Appears in: Unit 8: Applications of Integration