Math101 · Independent Ontario learning library
Math101Volumes by Cylindrical Shells
A rigorous, example-driven guide to volumes by cylindrical shells, including hypotheses, method choice, verification, and practice.
The central idea
A cylindrical shell formed by rotating a slice has volume $dV=2\pi(\text{radius})(\text{height})(\text{thickness})$. Vertical slices about a vertical axis typically give $V=2\pi\int r(x)h(x)dx$; horizontal slices about a horizontal axis give the analogous $dy$ formula. Radius is distance to the axis.
Definitions, hypotheses, and notation
When the axis passes through the region, naive shells from opposite sides may sweep the same radii and overlap. Reexpressing shell height as a function of radius or splitting carefully is necessary. Standard one-sided examples avoid this issue, but the geometry should always be checked.
Pappus's centroid theorem can verify some volumes when the axis does not intersect the region: volume equals area times distance traveled by the centroid. It is a check, not a replacement when centroid data are unavailable. Shell and washer methods must agree because both partition the same solid differently.
Conceptual meaning
A thin rectangle parallel to the rotation axis sweeps a hollow cylindrical layer. Its circumference times height times thickness approximates volume. Shells often avoid solving for inverse functions when washers would require horizontal slices.
A dependable method and decision rule
- Sketch the region and rotation axis.
- Choose slices parallel to the axis.
- Write radius as a nonnegative distance and height as top-minus-bottom or right-minus-left.
- Determine bounds along the slice-position variable.
- Integrate $2\pi rh$ and compare with a geometric scale estimate.
