Find the volume of revolution when the region bounded by the curve y=x^2+2 and y=6 is rotated about the y-axis.

I am confused when the region rotated is at both sides of the axis in which the region is rotated upon. If rotate one part of the region, the volume is

pi* integrate from 2 to 6 (y-2) dy.

Is it the same in this case, when the rotated region is at both sides? It seems when the region is rotated 2pi about the y-axis, it covers double the volume of rotating only one part of the region but if overlapping is to be considered, shouldn't both cases generate the same result?