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Volume by Slicing (Rotation about the x-axis) My Solutions> Recall that the volume of a solid formed by rotating a region about the x-axis can be found by slicing the solid perpendicular ta the x-axis and integrating dV, the volume of a single slice as shown in the figure below. (x.r(x) dx From the picture, we can see that the slice is a circular disk, so the volume is piR"2h, where (in this case) R is the function value at an arbitrary x and h-d. If there was a hole in the slice, the volume would be pi , R 2 , h-pi , m2h, where R is the outer function (farthest from the axis) and r is the inner function (closest to the axis). NOTE that if there is no hole, r 0 as we have the first formula again The general stratagy is similar to that for area: 1. Sketch the graph of the curve(s) 2. Find the points of intersection, if any 3, Determine the outer radius Rand inner radius r (if any) and integrate pi * 2-pi '2 over the desired interval Use this idea to find the volume of the solid formed by rotating the region bounded by y sin x y-0 x 0, and X-pi /2 about the x axis. Your Script a Save C Reset MATLAB Documentation symsx 2 f-g(x) 3a: ; % Define left boundary 4 b- ; % Define right boundary 5pafpl0t( ); % Plot over interval a to b 16|Rz ; % Define R 7 v-int( ) % integrate as described in the problem Run Script Assessment: Run Pretest Submit > Test for correct axes of plot (Pretest) Test for correct volume Test for correct radius Volume by Slicing (Rotation about the x-axis) My Solutions> Recall that the volume of a solid formed by rotating a region about the x-axis can be found by slicing the solid perpendicular ta the x-axis and integrating dV, the volume of a single slice as shown in the figure below. (x.r(x) dx From the picture, we can see that the slice is a circular disk, so the volume is piR"2h, where (in this case) R is the function value at an arbitrary x and h-d. If there was a hole in the slice, the volume would be pi , R 2 , h-pi , m2h, where R is the outer function (farthest from the axis) and r is the inner function (closest to the axis). NOTE that if there is no hole, r 0 as we have the first formula again The general stratagy is similar to that for area: 1. Sketch the graph of the curve(s) 2. Find the points of intersection, if any 3, Determine the outer radius Rand inner radius r (if any) and integrate pi * 2-pi '2 over the desired interval Use this idea to find the volume of the solid formed by rotating the region bounded by y sin x y-0 x 0, and X-pi /2 about the x axis. Your Script a Save C Reset MATLAB Documentation symsx 2 f-g(x) 3a: ; % Define left boundary 4 b- ; % Define right boundary 5pafpl0t( ); % Plot over interval a to b 16|Rz ; % Define R 7 v-int( ) % integrate as described in the problem Run Script Assessment: Run Pretest Submit > Test for correct axes of plot (Pretest) Test for correct volume Test for correct radius