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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 to the x-axis and integrating dV, the volume of a single slice as shown in the figure below. From the picture, we can see that the slice is a circular disk, so the volume is pi 2 h, where in this case) R is the function value at an arbitrary x and h = dx. If there was a hole in the slice, the volume would be pi R^2n-pi2 n , where is the outer function (farthest from the axis) and ris 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 strategy 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 R and inner radius r (if any) and integrate pi R'2 - pim2 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. Script Save C Reset DI MATLAB Documentation 1 syms x; 2 f=@(x); 3 = ; Define left boundary 4 b= ; Define right boundary 5 pafplot(); % Plot over interval a to b 6 R= ; Define R 7 Veint() % integrate as described in the problem Run Script 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 to the x-axis and integrating dV, the volume of a single slice as shown in the figure below. From the picture, we can see that the slice is a circular disk, so the volume is pi 2 h, where in this case) R is the function value at an arbitrary x and h = dx. If there was a hole in the slice, the volume would be pi R^2n-pi2 n , where is the outer function (farthest from the axis) and ris 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 strategy 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 R and inner radius r (if any) and integrate pi R'2 - pim2 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. Script Save C Reset DI MATLAB Documentation 1 syms x; 2 f=@(x); 3 = ; Define left boundary 4 b= ; Define right boundary 5 pafplot(); % Plot over interval a to b 6 R= ; Define R 7 Veint() % integrate as described in the problem Run Script