Shells Around an Axis Crossing the Region

Why a Crossing Axis Needs Care

A shell’s radius is a distance, so it cannot be negative. When the rotation axis lies outside a region, slices on one side of the axis have radii that change consistently as the slice moves across the region. When the axis passes through the region, slices can lie on either side, and their radii must be measured using absolute distance.

There is a second issue: a slice on one side of the axis and its reflected partner on the other side may generate the same cylindrical shell when rotated. Adding both contributions as if they were separate shells can count some volume twice. A correct setup uses each radius once and determines the shell’s height from the union of the slice segments at that radius.

Radius as Distance

Suppose a region is rotated about the vertical line \(x=a\). A vertical slice at coordinate \(x\) is at distance \(\lvert x-a\rvert\) from that axis. Its shell radius is therefore \(r=\lvert x-a\rvert\), not \(x-a\). The same principle applies to a horizontal slice rotated about \(y=b\): its radius is \(r=\lvert y-b\rvert\).

For example, if \(x<a\), then \(\lvert x-a\rvert=a-x\); if \(x>a\), then \(\lvert x-a\rvert=x-a\). Splitting at \(x=a\) expresses the distance correctly on each side. However, splitting the integral alone does not necessarily prevent double-counting: the shells formed at radius \(r\) by \(x=a-r\) and \(x=a+r\) are at the same distance from the axis.

Shell Height When Both Sides Contribute

For a vertical axis \(x=a\), consider a fixed radius \(r\ge 0\). The points in the original region that can generate a shell at that radius lie on the vertical lines \(x=a-r\) and \(x=a+r\), when those lines intersect the region. Each intersection gives one or more vertical segments. After rotation, segments at the same radius occupy the same cylindrical surface.

Thus the shell’s axial height is the total length of the union of those vertical segments, not necessarily the sum of their lengths. If both segments are identical, count their common length once. If they overlap only partly, count the overlap once and include any nonoverlapping parts. The shell formula is \(2\pi r\) times this height, multiplied by the thickness \(dr\).

where \(H(r)\) is the length of the union of the region’s slice segments at distance \(r\) from the axis. The integral is taken over radii that actually occur in the rotated region. For a horizontal axis, use horizontal slices and the corresponding union of horizontal segments.

Why the Radius and Union Rules Work

A thin slice at distance \(r\) from an axis sweeps out a cylindrical shell whose circumference is \(2\pi r\). If the slice has height \(H(r)\) and thickness \(dr\), the shell’s approximate volume is its circumference times its height times its thickness:

Adding these shell volumes over the radii gives the integral \(V=2\pi\int rH(r)\,dr\), under the usual conditions for the shell method: the region has well-defined slice lengths on the intervals being integrated, and the resulting height function is integrable. The distance \(r\) follows from the geometry: a radius is a length from the axis, so it is nonnegative. The union rule follows from the solid itself: when two slices create the same cylindrical surface, their common points are part of one solid, not two separate volumes. Integrating their heights as a sum would count those points twice.

Worked Example: A Rectangle Crossing a Vertical Axis

Worked Example: Unequal Slice Segments on Opposite Sides

Worked Example: Horizontal Shells Across the \(x\)-Axis

A Common Mistake: Signed Radii and Duplicate Shells

For the first rectangle, a tempting but incorrect setup is to use \(x\) as the radius and integrate over the original \(x\)-interval:

A negative volume is impossible. The error is using the signed coordinate \(x\) instead of the distance \(\lvert x\rvert\). Replacing \(x\) by \(\lvert x\rvert\) and integrating across the full interval does not, by itself, fix the setup:

This is still incorrect: the radii from \(x=-r\) and \(x=r\), where both slices exist, generate the same shells. The correct approach uses each radius once with its actual union height. In this example, that gives \(2\pi\int_0^2 3r\,dr=12\pi\).

Setting Up a Crossing-Axis Shell Integral

For a vertical axis \(x=a\), use vertical slices and proceed carefully from the axis outward. For a horizontal axis \(y=b\), use horizontal slices in the same way.

The key distinction is between the original coordinate and the shell radius. The coordinate can be negative; the radius cannot. And when both sides of an axis create shells at a given radius, the shell height describes the combined solid at that radius—not a sum of duplicate slices.