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

Common mistakes and why they fail