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Volumes by Shell
Volumes by Shells For some types of a solid, it is convenient to use hollow circular cylinders, or thin cylindrical shells, to calculate the volume of the solid of revolution. For Volume of a Solid, add up all the Volumes of each Shell Volume of the solid of revolution is:
Advantage of Shell Method over Disk/Washer Method We may use the Shell Method when we need to rotate the graph of a function of x- variable around the y-axis and vice versa. In this case, if we use the Disk Method, we have to transform the x-variable function into a function of y-variable, which is hard to do sometimes.
Example 1 The region bounded by the graph of y = 3x - x 2 and the x-axis is revolved about

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