Decision guideCalculus I · Unit 3BIntermediate

Washer method or shell method?

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

LaTeX article Updated July 11, 2026

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

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 example

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 these ideas

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