Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about the specified axis.

Question

x = 7y^2, y ≥ 0, x = 7; about y = 2

Answer 1

The region has vertices at
(0,0) (7,0) (7,1)

So, using shells of thickness dy, we have

v = ∫[0,1] 2πrh dy
where r = 2-y and h = 7-x = 7-7y^2
v = ∫[0,1] 2π(2-y)(7-7y^2) dy = 91π/6

Or, using discs (washers) of thickness dx,

v = ∫[0,7] π(R^2-r^2) dx
where R=2 and r=2-y=2-√(x/7)
v = ∫[0,7] 2π(2^2-(2-√(x/7))^2) dx = 91π/6