Question

Find the volume v of the solid obtained by rotating the region bounded by the given curves about the specified line. X = 9y2, x = 9; about x = 9.

Answer 1

To find the volume of the solid obtained by rotating the region bounded by the given curves about the line x = 9, we can use the method of cylindrical shells.

Step-by-step explanation:

To find the volume of the solid obtained by rotating the region bounded by the given curves about the line x = 9, we should use the method of cylindrical shells.

1. First, we need to determine the limits of integration, which will be the y-values at which the curves intersect. Setting x = 9y^2 and x = 9 equal to each other, we find that y = 1 and y = -1.
2. Next, we need to determine the height of each cylindrical shell. The height is given by the difference in y-values, which is 2 in this case.
3. We also need to determine the radius of each cylindrical shell. The radius is the distance from the axis of rotation (x = 9) to the curve at a given y-value. Since x = 9y^2, the radius is 9y^2 - 9.
4. Finally, we can set up the volume formula for cylindrical shells: V = 2π∫(9y^2 - 9)(2)dy, integrated from y = -1 to y = 1. Evaluating this integral will give us the volume of the solid.