Question

Suppose R is the region bounded by y = x^2, x = 2, and y = 0. A solid is generated by revolving R about the x = 3. Find the volume of the solid.

Answer 1

I believe the answer is 8pi, but I'd like to make sure.

Answer 2

A drawing would help you to visualize what's happening here. I'd start by graphing y=x^2, x=2 (a vertical line) and y=0 (the x-axis) and shading the area enclosed by them. Next, identify the line x=3.

Have you chosen one of the three methods available for finding the volumes of generated by rotating regions about a given axis? If so, which have you tried, and why?

Answer 3

To get 8pi, I set up two formulas to solve for volume. One with x=sqrt(y) on the interval of y=0 to 9. The second with x=sqrt(y) on the interval of y=4 to 9. Then I subtracted the second integral from the first.