The region between the graphs of y=x^2 and y=6x is rotated around the line x=8. The volume of the resulting solid is?

Answer 1

The curves intersect at (0,0) and (6,36). So, using discs (washers) of thickness dy,

v = ∫[0,36] π(R^2-r^2) dy
where R = 8-y/6 and r = 8-√y
v = ∫[0,36] π((8-y/6)^2-(8-√y)^2) dy = 360π

using shells of thickness dx,
v = ∫[0,6] 2πrh dx
where r=8-x and h=6x-x^2
v = ∫[0,6] 2π(8-x)(6x-x^2) dx = 360π