Chapter 6 | Applications of Integration
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The volume is 8 π units 3 .
6.9 Use the disk method to find the volume of the solid of revolution generated by rotating the region between the graph of g ( y ) = y and the y -axis over the interval [1, 4] around the y -axis.
The Washer Method Some solids of revolution have cavities in the middle; they are not solid all the way to the axis of revolution. Sometimes, this is just a result of the way the region of revolution is shaped with respect to the axis of revolution. In other cases, cavities arise when the region of revolution is defined as the region between the graphs of two functions. A third way this can happen is when an axis of revolution other than the x -axis or y -axis is selected. When the solid of revolution has a cavity in the middle, the slices used to approximate the volume are not disks, but washers (disks with holes in the center). For example, consider the region bounded above by the graph of the function f ( x ) = x and below by the graph of the function g ( x ) =1 over the interval [1, 4]. When this region is revolved around the x -axis, the result is a solid with a cavity in the middle, and the slices are washers. The graph of the function and a representative washer are shown in Figure 6.22 (a) and (b). The region of revolution and the resulting solid are shown in Figure 6.22 (c) and (d).
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