So we'll use the set-up that keeps things in terms of #x#. For instance if one boundary of the region is #y=x^4-8x^3+x+22# we do not want to try to rewrite this as a function of #y#. It is possible that you need to use a function whose inverse you can't find. The previous section introduced the Disk and Washer Methods, which computed the volume of solids of revolution by integrating the cross-sectional area of. Find the volume of the solid generated by revolving the region about the x-axis. If you set it up one way and don't care for the looks of the integral, try setting it up the other way. Let R be a region in the 1st quadrant enclosed by y 4-x 2, y 3x, and the y-axis. y-axis, instead of a disk or a washer, we get a cylindrical shell, as shown in the. As a student, perhaps because we learned disks/washers first, I preferred them. Figure 2.25 (a) A region bounded by the graph of a function of x. Taking #y=0#, #y=x^2#, and #y=-x+2# around the #x#-axis, I would use shells to avoid doing two integrals even though it would require me to rewrite the curves as functions of #y#.Īgain, that is my preference. (Disks would require two: one from #y=0# to #y=1# and another from #y=1# to #y=2#.) So if I have to find the volume of the solid generated by revolving the region bounded by #x=0#, #y=x^2#, and #y=-x+2# around the #y#-axis, I would use shells because there would only be one integral to evaluate. I have a preference for doing a single integral. Scale your graph so that the bounded region takes up a significant portion of the page. Sometimes one leads to an integral that a particular person finds easier to evaluate, but what is easier varies between people. Disk/Washer Method On a sheet of graph paper, carefully graph the region bounded by x 0, y 0, y 4 - 2x, rotated about the line x -1.
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