The problem asks to find the volume of the region bounded between y=x^2 and y=2-x, with the region being rotated around the vertical line x=3.

This problem seems rather complex, and I have dwelled on it longer than I care to admit. Because it is rotated around the vertical axis, I gather that I must slice horizontally and integrate with respect to y, but I can't seem to get any further.

My confusion is further exacerbated by the fact that, assuming integration with respect to y, when y is between 0 and 1, the inner and outter bounds of the shape are dictated solely by the line y=x^2.

Is there some way that I am overlooking to integrate with respect to x? I am so confused!

Thank you in advance for any assistance.