Decision guideCalculus IIIntermediate11 min read

Washer method or shell method?

Choose slices by geometry: perpendicular slices create washers, parallel slices create cylindrical shells.

V=\pi\int(R^2-r^2)\,d\!x\quad\text{or}\quad V=2\pi\int(\text{radius})(\text{height})\,d\!x

Decision guide

A practical comparison that turns a vague choice into a repeatable test.

Reviewed July 11, 2026

DecisionStart here

Use the setup that keeps the region in one piece and expresses its dimensions with the least algebra. The axis orientation determines whether dx or dy is natural.

What each slice becomes

A slice perpendicular to the axis sweeps out a disk or washer, so its cross-sectional area is π(R² − r²). A slice parallel to the axis sweeps out a thin shell with circumference, height, and thickness.

Both methods compute the same volume from different decompositions.

Predict the easier variable

Sketch a representative slice before writing any formula. If vertical slices force you to solve for x in several branches, horizontal shells or washers may be cleaner.

Avoid a method that requires splitting the region unless the alternative is worse.

Radii are distances

A radius is the distance from the slice to the axis of rotation, not automatically x or y. Shifted axes require expressions such as x + 2 or 5 − y.

Worked exampleShells avoid an inverse

Rotate the region under y = x² from x = 0 to 1 about the y-axis.

r=x,\qquad h=x^2

A vertical slice is parallel to the y-axis and forms a shell.

V=2\pi\int_0^1x(x^2)\,dx

Circumference times height times thickness.

V=2\pi\left[\frac{x^4}{4}\right]_0^1

Evaluate the simple polynomial integral.

Result

\boxed{V=\frac\pi2}

Watch for

Common mistakes

Using x as the radius when the axis is shifted.
Mixing dx with dimensions written as functions of y.
Forgetting the inner radius in a washer.

Keep

Three takeaways

Draw the slice and its rotated shape.
Perpendicular means washer; parallel means shell.
Choose the method that avoids splitting or inverses.