![]() The shell is a cylinder, so its volume is the cross-sectional area multiplied by the height of the cylinder. ![]() To calculate the volume of this shell, consider. 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. Then, construct a rectangle over the interval Ī representative rectangle is shown in (a). Using a regular partition, P = Ĭhoose a point x i * ∈. Previously, regions defined in terms of functions of xĪs we have done many times before, partition the interval Note that this is different from what we have done before. We then revolve this region around the y-axis, as shown in (b). As before, we define a region R ,īounded above by the graph of a function y = f ( x ) ,Īnd on the left and right by the lines x = a The Method of Cylindrical ShellsĪgain, we are working with a solid of revolution. 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. ![]() Also, the specific geometry of the solid sometimes makes the method of using cylindrical shells more appealing than using the washer method. The ability to choose which variable of integration we want to use can be a significant advantage with more complicated functions. With the method of cylindrical shells, we integrate along the coordinate axis perpendicular to the axis 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. In this section, we examine the method of cylindrical shells, the final method for finding the volume of a solid of revolution.
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