Well lets look at it like this. We have a parabola bounded by the vertical lines x=0, x=4 and the horizontal line y=0

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?