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  • Calculate the volume of a solid of revolution by using the method of cylindrical shells.
  • Compare the different methods for calculating a volume of revolution.

In this section, we examine the method of cylindrical shells, the final method for finding the volume of a solid of revolution. We can use this method on the same kinds of solids as the disk method or the washer method; however, with the disk and washer methods, we integrate along the coordinate axis parallel to the axis of revolution. With the method of cylindrical shells, we integrate along the coordinate axis perpendicular to the axis of revolution. The ability to choose which variable of integration we want to use can be a significant advantage with more complicated functions. Also, the specific geometry of the solid sometimes makes the method of using cylindrical shells more appealing than using the washer method. In the last part of this section, we review all the methods for finding volume that we have studied and lay out some guidelines to help you determine which method to use in a given situation.

The method of cylindrical shells

Again, we are working with a solid of revolution. As before, we define a region R , bounded above by the graph of a function y = f ( x ) , below by the x -axis, and on the left and right by the lines x = a and x = b , respectively, as shown in [link] (a). We then revolve this region around the y -axis, as shown in [link] (b). Note that this is different from what we have done before. Previously, regions defined in terms of functions of x were revolved around the x -axis or a line parallel to it.

This figure has two graphs. The first graph is labeled “a” and is an increasing curve in the first quadrant. The curve is labeled “y=f(x)”. The curve starts on the y-axis at y=a. Under the curve, above the x-axis is a shaded region labeled “R”. The shaded region is bounded on the right by the line x=b. The second graph is a three dimensional solid. It has been created by rotating the shaded region from “a” around the y-axis.
(a) A region bounded by the graph of a function of x . (b) The solid of revolution formed when the region is revolved around the y -axis .

As we have done many times before, partition the interval [ a , b ] using a regular partition, P = { x 0 , x 1 ,… , x n } and, for i = 1 , 2 ,… , n , choose a point x i * [ x i 1 , x i ] . Then, construct a rectangle over the interval [ x i 1 , x i ] of height f ( x i * ) and width Δ x . A representative rectangle is shown in [link] (a). When that rectangle is revolved around the y -axis, instead of a disk or a washer, we get a cylindrical shell, as shown in the following figure.

This figure has two images. The first is a cylindrical shell, hollow in the middle. It has a vertical axis in the center. There is also a curve that meets the top of the cylinder. The second image is a set of concentric cylinders, one inside of the other forming a nesting of cylinders.
(a) A representative rectangle. (b) When this rectangle is revolved around the y -axis , the result is a cylindrical shell. (c) When we put all the shells together, we get an approximation of the original solid.

To calculate the volume of this shell, consider [link] .

This figure is a graph in the first quadrant. The curve is increasing and labeled “y=f(x)”. The curve starts on the y-axis at f(x*). Below the curve is a shaded rectangle. The rectangle starts on the x-axis. The width of the rectangle is delta x. The two sides of the rectangle are labeled “xsub(i-1)” and “xsubi”.
Calculating the volume of the shell.

The shell is a cylinder, so its volume is the cross-sectional area multiplied by the height of the cylinder. The cross-sections are annuli (ring-shaped regions—essentially, circles with a hole in the center), with outer radius x i and inner radius x i 1 . Thus, the cross-sectional area is π x i 2 π x i 1 2 . The height of the cylinder is f ( x i * ) . Then the volume of the shell is

V shell = f ( x i * ) ( π x i 2 π x i 1 2 ) = π f ( x i * ) ( x i 2 x i 1 2 ) = π f ( x i * ) ( x i + x i 1 ) ( x i x i 1 ) = 2 π f ( x i * ) ( x i + x i 1 2 ) ( x i x i 1 ) .

Note that x i x i 1 = Δ x , so we have

V shell = 2 π f ( x i * ) ( x i + x i 1 2 ) Δ x .

Furthermore, x i + x i 1 2 is both the midpoint of the interval [ x i 1 , x i ] and the average radius of the shell, and we can approximate this by x i * . We then have

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Source:  OpenStax, Calculus volume 2. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11965/1.2
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