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Calculus IIUniversity3 min read

Volumes by Cylindrical Shells

A rigorous, example-driven guide to volumes by cylindrical shells, including hypotheses, method choice, verification, and practice.

Cheat sheet

A multiple integral accumulates small area or volume elements over a region. A translucent solid is divided into representative elliptical slices. The repeated cross-sections show how a volume can be accumulated one thin slice at a time.

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.

Fully worked example

Graphical or geometric meaning

At each $x$, rotation creates a shell of circumference $2\pi x$ and height $x$. Small-radius shells are short; large-radius shells are tall, so outer layers dominate the volume.

Common mistakes and why they fail

Verification and reasonableness checks

Confirm radius, height, and thickness have length units.
Compare with a washer setup or known solid when possible.
Ensure shells cover the solid once without gaps or overlap.

A shell is circumference times height

A shell parallel to the rotation axis contributes approximately $2\pi(\text{radius})(\text{height})(\text{thickness})$. Radius is the nonnegative distance to the axis; height is top minus bottom for vertical shells or right minus left for horizontal shells. Sketch one shell and express both dimensions using one variable. Shells are efficient when washers require solving for another variable or splitting, but either sound method should agree. If the region crosses the axis, ensure the swept volume is not counted twice. The answer needs cubic units and should lie below a simple enclosing-cylinder estimate. These geometric checks are often faster than redoing a long antiderivative.

Practice

Rotate $0\le y\le1$, $0\le x\le3$ about the $y$-axis by shells.
What is shell radius about $x=5$ for a slice at $x$?
What orientation are shell slices relative to the axis?

Answers and brief solutions

$9\pi$.
$|5-x|$.
Parallel.

Connections and next steps

Check your understanding

Try it yourself

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1 practice question

Question 1Set up cylindrical shells · Standard

What is the shell-method volume when the region under y=x on [0,1] rotates about the y-axis?

End of lesson

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