Identifying Shell Radius, Height, and Thickness

From a Slice to a Shell

A cylindrical shell begins as a thin slice of a region. When that slice is rotated around an axis parallel to it, the slice sweeps out a thin, hollow cylindrical layer. To describe the layer correctly, identify three dimensions: its radius, its height, and its thickness.

These dimensions come from different geometric measurements. The radius measures how far the slice is from the axis of rotation. The height measures the length of the slice within the region. The thickness measures how far one representative slice is from the next. Once those roles are clear, the shell’s small volume is approximately its circumference times its height times its thickness.

The Three Dimensions

Consider a region in the \(xy\)-plane and a chosen axis of rotation. A representative slice is a narrow rectangle inside the region, parallel to that axis. As the slice rotates, it traces a shell.

• Radius, \(r\): the perpendicular distance from the representative slice to the axis of rotation. A radius is a nonnegative distance, not a signed coordinate.
• Height, \(h\): the length of the representative slice that lies inside the region. It is measured parallel to the axis of rotation.
• Thickness: the small width of the slice, measured perpendicular to the slice and in the direction that moves the slice through the region. For vertical slices this is usually \(dx\); for horizontal slices it is usually \(dy\).

For example, if vertical slices move across a region as \(x\) changes, the shell thickness is \(dx\). If horizontal slices move across a region as \(y\) changes, the thickness is \(dy\). The thickness is not the shell’s radius and is not its height.

Why the Formula Uses These Dimensions

A thin shell has circumference approximately \(2\pi r\). If its height is \(h\), unrolling a very thin shell gives a nearly rectangular strip: its length is approximately \(2\pi r\), its height is \(h\), and its thickness is the small width \(dq\). The strip’s volume is therefore approximately

The approximation becomes exact in the limiting process used to form the integral. The representative dimensions vary from slice to slice, so the total volume is accumulated by integrating the shell-volume elements. The earlier derivation of the cylindrical shell volume formula establishes this limiting argument; the practical task here is to identify the correct geometric quantities before using it.

Units provide a useful check. If \(r\), \(h\), and \(dq\) are measured in centimeters, then \(2\pi rh\,dq\) has units of cubic centimeters, as a volume should. If the result has units of area or length, a dimension has been omitted or misinterpreted.

Worked Example: Vertical Slice Rotated About the \(y\)-Axis

Worked Example: Horizontal Slice Rotated About the \(x\)-Axis

Worked Example: A Slice and an Axis on the Same Coordinate Grid

A Reliable Identification Process

Before forming a shell-volume expression, answer three separate questions. Keeping the questions distinct helps prevent confusing a coordinate, a slice length, and a slice width.

When the dimensions depend on position, write them as functions of the slice variable. For example, a vertical slice might have radius \(r(x)\), height \(h(x)\), and thickness \(dx\), giving \(dV=2\pi r(x)h(x)\,dx\). This notation keeps the geometry visible in the expression rather than hiding it inside a memorized formula.

Common Mistake: Confusing Labels with a Changed Product

Radius and height describe different aspects of the shell, so swapping their labels gives a geometrically incorrect description. However, the swap alone does not change the shell-volume formula: multiplication is commutative. In the first example, calling \(4\) the radius and \(2\) the height would still give

This equality does not make the labels correct; the actual radius is \(2\), and the actual height is \(4\). It does show why a mislabeled radius and height, by itself, cannot change the product \(2\pi rh\). A wrong volume expression occurs if the confusion leads to an incorrect dimension or incorrect bounds.

For instance, in that same example the radius is the distance from \(x=2\) to the \(y\)-axis \(x=0\), so \(r=2\). Using the slice’s top coordinate \(y=5\) as the radius would instead produce the incorrect expression \(2\pi(5)(4)\,dx=40\pi\,dx\). The correction is to measure radius perpendicular to the axis, not to choose an endpoint coordinate unrelated to that distance.