Volumes by Disks and Washers

The central idea

Slices perpendicular to a rotation axis produce disks or washers. A washer with outer radius $R$ and inner radius $r$ has cross-sectional area $A=\pi(R^2-r^2)$, so $V=\int A\,dx$ or $\int A\,dy$. Radii are nonnegative distances from the axis, and bounds run perpendicular to the slices.

Definitions, hypotheses, and notation

A shifted axis changes both radii. Rotating the region between $y=f(x)$ and $y=g(x)$ about $y=k$ requires distances $|f-k|$ and $|g-k|$, ordered geometrically rather than by which function is larger. If the axis crosses a slice, the inner radius may become zero and a split can be necessary.

Cavalieri's principle says solids with equal cross-sectional areas at every position have equal volume, explaining why the method does not require a named three-dimensional shape. Shells and washers are complementary partitions; choose the one that avoids inverses and excessive pieces.

Conceptual meaning

Rotating one segment perpendicular to the axis produces a circular disk; if the segment begins away from the axis, it produces a washer with a hole. Stacking thin cross-sections reconstructs the solid.

A dependable method and decision rule

1. Sketch the region and axis of rotation.
2. Choose slices perpendicular to the axis.
3. Identify outer and inner distances from the axis.
4. Write $\pi(R^2-r^2)$, splitting if boundary roles change.
5. Integrate over the slice-position bounds and check cubic units.

Fully worked example

Graphical or geometric meaning

A stack of disks grows in radius as $x$ increases. Squaring radius converts each slice's boundary distance into cross-sectional area; thickness then turns area into a small volume.

Common mistakes and why they fail

Verification and reasonableness checks

• Verify $R\ge r\ge0$ throughout each interval.
• Compare against a shell-method setup.
• Use bounding cylinders to estimate volume.

Cross-sections are perpendicular to the axis

A washer has area $\pi(R^2-r^2)$, where each radius is a nonnegative distance from the axis. Use slices perpendicular to that axis: vertical for a horizontal axis and horizontal for a vertical axis. Determine outer and inner radii geometrically, accounting for any shifted axis, and split where their roles change. A disk is the case $r=0$. If the region crosses the axis, describe the actual swept cross-section so overlapping rotations are not counted twice. The integral must have cubic units and a nonnegative value. An enclosing cylinder gives an immediate upper-bound check, while a sketch confirms that no unintended gap or overlap has entered the setup.

Practice

1. Rotate $y=2$, $0\le x\le3$ about the $x$-axis. Find volume.
2. A washer has radii 5 and 3. Find its area.
3. What slice orientation produces washers?

Connections and next steps

• Volumes by Cylindrical Shells
• Area Between Curves
• Applications of Integration