Shell integration

Shell integration (the shell method in integral calculus) is a means of calculating the volume of a solid of revolution, when integrating along an axis perpendicular to the axis of revolution.

It makes use of the so-called "representative cylinder". Intuitively speaking, part of the graph of a function is rotated around an axis, and is modelled by an infinite number of hollow pipes, all infinitely thin.

The idea is that a "representative rectangle" (used in the most basic forms of integration – such as ∫ x dx) can be rotated about the axis of revolution; thus generating a hollow cylinder. Integration, as an accumulative process, can then calculate the integrated volume of a "family" of shells (a shell being the outer edge of a hollow cylinder) – as volume is the antiderivative of area, if one can calculate the lateral surface area of a shell, one can then calculate its volume.

Shell integration can be considered a special case of evaluating a double integral in polar coordinates.

Calculation

For a solid formed by revolution of a region around the y-axis, the volume may be calculated as

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

where h(x) is the height of the rotated region at distance x from the y-axis.

See also

• Solid of revolution
• Disk integration
• Radius

References

• CliffsNotes.com. Volumes of Solids of Revolution. 12 Apr 2011 <http://www.cliffsnotes.com/study_guide/topicArticleId-39909,articleId-39907.html>.
• Weisstein, Eric W. "Method of Shells". MathWorld.
• Frank Ayres, Elliott Mendelson:Schaum's outlines: Calculus. McGraw-Hill Professional 2008, ISBN 978-0-07-150861-2. pp. 244–248 (online copy, p. 244, at Google Books)