![\"This](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11213029\/CNX_Calc_Figure_06_03_009.jpg\")
Figure 10.[\/caption]\n\nWe have studied several methods for finding the volume of a solid of revolution, but how do we know which method to use? It often comes down to a choice of which integral is easiest to evaluate.<\/p>\n
The figure below describes the different approaches for solids of revolution around the [latex]x\\text{-axis}.[\/latex] It\u2019s up to you to develop the analogous table for solids of revolution around the [latex]y\\text{-axis}.[\/latex]<\/p>\n\n[caption id=\"\" align=\"aligncenter\" width=\"974\"] Figure 10.[\/caption]\n\n
Let\u2019s take a look at a couple of additional problems and decide on the best approach to take for solving them.<\/p>\n The tips below can help you decide the best method:<\/p>\n Most times, functions are presented in terms of [latex]x[\/latex]. If possible, keeping things in terms of [latex]x[\/latex] is beneficial.<\/p>\n Generally speaking, for an [latex]x[\/latex]-axis revolution, a disk\/washer method will allow us to avoid rewriting the equation in terms of [latex]y[\/latex]. For a [latex]y[\/latex]-axis revolution, the shell method will allow us the same advantage.<\/p>\n Using this information, try to draw your rectangles in terms of [latex]dx[\/latex], if possible. If this requires you to separate the area, try the [latex]dy[\/latex] method!<\/p>\n<\/section>\n For each of the following problems, select the best method to find the volume of a solid of revolution generated by revolving the given region around the [latex]x\\text{-axis},[\/latex] and set up the integral to find the volume (do not evaluate the integral).<\/p>\n Looking at the region, if we want to integrate with respect to [latex]x,[\/latex] we would have to break the integral into two pieces, because we have different functions bounding the region over [latex]\\left[0,1\\right][\/latex] and [latex]\\left[1,2\\right].[\/latex]<\/p>\n In this case, using the disk method, we would have:<\/p>\n If we used the shell method instead, we would use functions of [latex]y[\/latex] to represent the curves, producing:<\/p>\n Neither of these integrals is particularly onerous, but since the shell method requires only one integral, and the integrand requires less simplification, we should probably go with the shell method in this case.<\/p>\n<\/li>\n\t Looking at the region, it would be problematic to define a horizontal rectangle; the region is bounded on the left and right by the same function. Therefore, we can dismiss the method of shells. The solid has no cavity in the middle, so we can use the method of disks. Then:<\/p>\n [\/hidden-answer]<\/p>\n<\/div>\n<\/section>\n Select the best method to find the volume of a solid of revolution generated by revolving the given region around the [latex]x\\text{-axis},[\/latex] and set up the integral to find the volume (do not evaluate the integral): the region bounded by the graphs of [latex]y=2-{x}^{2}[\/latex] and [latex]y={x}^{2}.[\/latex]<\/p>\n Use the method of washers<\/p>\n [latex]V={\\displaystyle\\int }_{-1}^{1}\\pi \\left[{(2-{x}^{2})}^{2}-{({x}^{2})}^{2}\\right]dx[\/latex]<\/p>\n [\/hidden-answer]<\/p>\n<\/div>\n<\/div>\n <\/p>\n<\/section>\n Watch the following video to see the worked solution to the two examples above.<\/p>\n\n\n
\n Axis of Revolution<\/th>\n Method<\/th>\n Variable<\/th>\n Description<\/th>\n<\/tr>\n \n [latex]x[\/latex]-axis<\/td>\n Disk\/washer<\/td>\n [latex]dx[\/latex]<\/td>\n Use when revolving around the [latex]x[\/latex]-axis and integrating with respect to [latex]x[\/latex]. The rectangles are perpendicular to the [latex]x[\/latex]-axis.<\/td>\n<\/tr>\n \n Shell<\/td>\n [latex]dy[\/latex]<\/td>\n Use when revolving around the [latex]x[\/latex]-axis and integrating with respect to [latex]y[\/latex]. The rectangles are parallel to the [latex]x[\/latex]-axis.<\/td>\n<\/tr>\n \n [latex]y[\/latex]-axis<\/td>\n Disk\/washer<\/td>\n [latex]dy[\/latex]<\/td>\n Use when revolving around the [latex]y[\/latex]-axis and integrating with respect to [latex]y[\/latex]. The rectangles are perpendicular to the [latex]y[\/latex]-axis.<\/td>\n<\/tr>\n \n Shell<\/td>\n [latex]dx[\/latex]<\/td>\n Use when revolving around the [latex]y[\/latex]-axis and integrating with respect to [latex]x[\/latex]. The rectangles are parallel to the [latex]y[\/latex]-axis.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n \n\t
[reveal-answer q=\"fs-id1167793244629\"]Show Solution[\/reveal-answer]
\n[hidden-answer a=\"fs-id1167793244629\"]\n\n\n\t
\n\n[caption id=\"\" align=\"aligncenter\" width=\"708\"]![\"This](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11213032\/CNX_Calc_Figure_06_03_010.jpg\")
Figure 11. (a) The region [latex]R[\/latex] bounded by two lines and the [latex]x\\text{-axis}.[\/latex] (b) The solid of revolution generated by revolving [latex]R[\/latex] about the [latex]x\\text{-axis}.[\/latex][\/caption]\n<\/div>\n <\/div>\n[latex]V={\\displaystyle\\int }_{0}^{1}(\\pi {x}^{2})dx+{\\displaystyle\\int }_{1}^{2}(\\pi {(2-x)}^{2})dx.[\/latex]<\/div>\n[latex]\\begin{array}{cc}\\hfill V& ={\\displaystyle\\int }_{0}^{1}(2\\pi y\\left[(2-y)-y\\right])dy\\hfill \\\\ & ={\\displaystyle\\int }_{0}^{1}(2\\pi y\\left[2-2y\\right])dy.\\hfill \\end{array}[\/latex]<\/div>\n
\n\n[caption id=\"\" align=\"aligncenter\" width=\"709\"]![\"This](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11213036\/CNX_Calc_Figure_06_03_011.jpg\")
Figure 12. (a) The region [latex]R[\/latex] between the curve and the [latex]x\\text{-axis}.[\/latex] (b) The solid of revolution generated by revolving [latex]R[\/latex] about the [latex]x\\text{-axis}.[\/latex][\/caption]\n<\/div>\n <\/div>\n[latex]V={\\displaystyle\\int }_{0}^{4}\\pi {(4x-{x}^{2})}^{2}dx.[\/latex]<\/div>\n<\/li>\n<\/ol>\n\n
\n[reveal-answer q=\"967053\"]Hint[\/reveal-answer]
\n[hidden-answer a=\"967053\"]Sketch the region and use the last example to decide which integral is easiest to evaluate.[\/hidden-answer]<\/p>\n[reveal-answer q=\"fs-id1167793496106\"]Show Solution[\/reveal-answer]
\n[hidden-answer a=\"fs-id1167793496106\"]\n\n