So this would appear to be best for the method of cylindrical shells.
The interior of the bowl corresponds to the revolving region given by
The area element at position x has height
and generates the clindrical shell of volume
Edit- Was it just my imagination or didn't I see a moderator post in this thread? They had something different then me but I don't see why what I've done would be wrong. Although it has been a while since I did cylindrical shells. Can you post again?