Shell integration
Part of a series of articles about  
Calculus  



Specialized 

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. This is in contrast to disk 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 crosssection in the xyplane around the yaxis. Suppose the crosssection is defined by the graph of the positive function f(x) on the interval [a, b]. Then the formula for the volume will be:
If the function is of the y coordinate and the axis of rotation is the xaxis then the formula becomes:
If the function is rotating around the line x = h or y = k, the formulas become:^{[1]}
and
The formula is derived by computing the double integral in polar coordinates.
Example
Consider the volume, depicted below, whose cross section on the interval [1, 2] is defined by:
In the case of disk integration we would need to solve for x given y. Because the volume is hollow in the middle we will find two functions, one that defines the inner solid and one that defines the outer solid. After integrating these two functions with the disk method we subtract them to yield the desired volume.
With the shell method all we need is the following formula:
By expanding the polynomial the integral becomes very simple. In the end we find the volume is π/10 cubic units.
See also
References
 ↑ Heckman, Dave (2014). "Volume – Shell Method" (PDF). Retrieved 20160928.
 Frank Ayres, Elliott Mendelson. Schaum's Outlines: Calculus. McGrawHill Professional 2008, ISBN 9780071508612. pp. 244–248 (online copy, p. 244, at Google Books)