Application of Integration:
Solids of Revolution by the Shell Method

In addition to the disc or washer method of finding volumes of revolution, a popular method is the Shell Method. this is particularly useful when the area defined revolved around the vertical y-axis and the region is a function in terms of x that is difficult (or impossible) to solve for y.

The basic formula arises out of building cylindrical shells based on infinitely thin rectangles revolved around an axis. These rectangles sweep out shells and if we cut them into strips these strips are "box-like" ... infinitely thin but with length and height. The volume of these strips are the product of their length, height and width. The length is the circumference of the circular shell.

C = 2p r

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Axis of Revolution is Vertical:

The formula for the volume of such a solid is:

where p(x) is the distance of the representative rectangle (in orange) from the x-axis (often this is simply x) and h(x) is the height of the rectangle (often simply the y-value of the function).

An Example:

The region bounded by the graph of on the interval [0, 1] and the x-axis is revolved around the y-axis.

Note that the region being rotated is only the region in the first quadrant on the interval [0, 1].

Here p(x) = x and , yielding

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Axis of Revolution is Horizontal:

These problems often arise when presented with a function of x in terms of y. The formula for the volume of such a solid is:

where p(y) is the distance of the representative rectangle (in orange) from the x-axis (often this is simply y) and h(y) is the width of the rectangle (often simply the x-value of the function).

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