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Shell integration

Method for calculating the volume of a solid of revolution

Wikipedia contributorsAdapted for Fathomly · View authors
1 min read
Source updated June 26, 2026
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Shell integration (the shell method in integral calculus) is a method for calculating the volume of a solid of revolution, when integrating along an axis perpendicular to the axis of revolution. This is in contrast to disc integration which integrates along the axis parallel to the axis of revolution.

01Definition

The shell method goes as follows: Consider a volume in three dimensions obtained by rotating a cross-section in the xy-plane around the y-axis. Suppose the cross-section is defined by the graph of the positive function f(x) on the interval [a, b]. Then the formula for the volume will be:

2\pi \int _{a}^{b}xf(x)\,dx

If the function is of the y coordinate and the axis of rotation is the x-axis then the formula becomes:

2\pi \int _{a}^{b}yf(y)\,dy

If the function is rotating around the line x = h then the formula becomes:

{\begin{cases}\displaystyle 2\pi \int _{a}^{b}(x-h)f(x)\,dx,&{\text{if}}\ h\leq a<b\\\displaystyle 2\pi \int _{a}^{b}(h-x)f(x)\,dx,&{\text{if}}\ a<b\leq h,\end{cases}}

and for rotations around y = k it becomes

{\begin{cases}\displaystyle 2\pi \int _{a}^{b}(y-k)f(y)\,dy,&{\text{if}}\ k\leq a<b\\\displaystyle 2\pi \int _{a}^{b}(k-y)f(y)\,dy,&{\text{if}}\ a<b\leq k.\end{cases}}

The formula is derived by computing the double integral in polar coordinates.

Derivation of the formula

A volume is approximated by a collection of hollow cylinders. As the cylinder walls get thinner the approximation gets better. The limit of this approximation is the shell integral.
A volume is approximated by a collection of hollow cylinders. As the cylinder walls get thinner the approximation gets better. The limit of this approximation is the shell integral.

02Example

Consider the volume, depicted below, whose cross section on the interval [1, 2] is defined by:

y=8(x-1)^{2}(x-2)^{2}

With the shell method we simply use the following formula:

V=16\pi \int _{1}^{2}x((x-1)^{2}(x-2)^{2})\,dx

By expanding the polynomial, the integration is easily done giving ⁠8/10⁠\pi cubic units.

Comparison with disc integration

Much more work is needed to find the volume if we use disc integration. First, we would need to solve y=8(x-1)^{2}(x-2)^{2} for x. Next, because the volume is hollow in the middle, we would need two functions: one that defined an outer solid and one that defined the inner hollow. After integrating each of these two functions, we would subtract them to yield the desired volume.

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Sources and credits

This article is adapted from the Wikipedia article “Shell integration”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Related topics

Solid of revolution

In geometry, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line, which may not intersect the generatrix (except at its boundary). The surface created by this revolution and which bounds the solid is the surface of revolution.

Disc integration

Disc integration, also known in integral calculus as the disc method, is a method for calculating the volume of a solid of revolution of a solid-state material when integrating along an axis parallel to the axis of revolution. This method models the resulting three-dimensional shape as a stack of an infinite number of discs of varying radius and infinitesimal thickness.