Cylindrical shell volume
Build a solid of revolution from thin cylindrical shells.
Cylindrical shell volume is one of 7 applications formulas in the calculus section of this library, and it is used at ap · university level.
Why cylindrical shell volume works
Cut the region into thin strips running parallel to the rotation axis. Spinning one strip sweeps out a hollow tube, and slitting that tube open flattens it into a rectangle whose length is the circle the strip travelled, whose height is the strip height, and whose thickness is the strip width.
What each symbol means
Radius is distance to the axis; height is top minus bottom.
Cylindrical shell volume: when it holds
Use slices parallel to the axis and nonnegative geometric radius and height.
When it stops applying
Each strip must be parallel to the axis and lie wholly on one side of it. If the region straddles the rotation axis, the two halves sweep through the same space and integrating across both counts that solid twice.
Cylindrical shell volume: a worked example
Rotating the region under $y=x$ on $[0,1]$ about the $y$-axis gives $V=2\pi\int_0^1x^2dx=2\pi/3$.
The mistake to avoid
What people do: Dropping the 2π and integrating only the radius times the height.
Why it goes wrong: That produces an area rather than a volume, because the circumference each shell sweeps through has been thrown away entirely.
Do this instead: Keep the 2π factor outside the integral and multiply radius by height inside it.
Cylindrical shell volume: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. Radius is distance to the axis; height is top minus bottom.
- Check the conditions before substituting. Use slices parallel to the axis and nonnegative geometric radius and height.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most calculus slips.
Where this formula fits
- Subject
- Calculus formulas — 42 entries in this library
- Topic
- Applications
- Level
- AP · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where cylindrical shell volume comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Fundamental Theorem of Calculus — the lesson behind this formula: connect accumulated area with instantaneous change.
- Calculus Calculator — check your substitution and the value it produces.
- Study calculus — the subject guide that explains the ideas these formulas compress.
- Calculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 42 calculus formulas — the full grouped reference, or the complete formula library.
Questions about cylindrical shell volume
When are shells easier than washers?
When rewriting the curve in terms of the other variable would be painful. A cubic like y = x cubed plus x spun about the vertical axis is far easier with shells.
What is the radius when the axis is the vertical one?
It is the horizontal coordinate itself, since that is the distance from the strip out to the axis, and the integrand becomes 2π times x times the height.
Why do shells and washers agree on the same solid?
They cut the identical object in two different directions, so only the bookkeeping differs. Any disagreement between them signals an algebra slip somewhere.
How do I work out the height of a shell?
Subtract the lower boundary from the upper boundary at that position, in exactly the way you would set up an area between two curves.