Multiple choice 1 point
Find the volume of the solid obtained by rotating the region bounded by and the axis about the axis.



None of these

To find the volume of the solid obtained by rotating the region bounded by , , and the -axis about the -axis, we start by determining the intersection point of and . Setting gives .

We use the method of cylindrical shells, where the volume is given by the integral:

Now, substitute , changing the limits accordingly from to :

Calculating the integral:

The total volume is then:

However, upon re-evaluating our approach carefully, we find that the rotation process involves boundaries that expand upon being reflected about the -axis leading us to examine fractions of the whole shape. After careful integration again or inspecting for the correct volume formulations, we notice that another formulation for revolutions also may provide distinct integration values. It can be determined through the cross-sectional area or verifying volume components that either redirection would ascertain as truly established.

Therefore, the answer is .