Shell integration

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Method for calculating the volume of a solid of revolution

title: "Shell integration" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["integral-calculus"] description: "Method for calculating the volume of a solid of revolution" topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Shell_integration" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Method for calculating the volume of a solid of revolution ::

::figure[src="https://upload.wikimedia.org/wikipedia/commons/3/31/Shell_integral_undergraph_-_around_y-axis.png" caption="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."] ::

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.

Definition

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 x f(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 y f(y), dy

If the function is rotating around the line then the formula becomes:

:\begin{cases} \displaystyle 2 \pi \int_a^b (x-h) f(x),dx, & \text{if}\ h \le a \displaystyle 2 \pi \int_a^b (h-x) f(x),dx, & \text{if}\ a \end{cases} and for rotations around it becomes :\begin{cases} \displaystyle 2 \pi \int_a^b (y-k) f(y),dy, & \text{if}\ k \le a \displaystyle 2 \pi \int_a^b (k-y) f(y),dy, & \text{if}\ a \end{cases}

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

Derivation of the formula

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The method's formula can be derived as follows:
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Example

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 \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.

References

Frank Ayres, Elliott Mendelson. Schaum's Outlines: Calculus. McGraw-Hill Professional 2008, . pp. 244–248 ()

References

Heckman, Dave. (2014). "Volume – Shell Method".

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