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. x = 7y^2, y ≥ 0, x = 7; about y = 2

Answer 1

the curves intersect at (7,1) and (7,-1)
Using shells of thickness dy, we have
v = ∫[-1,1] 2πrh dy
where r = 2-y and h = 7-7y^2
v = ∫[-1,1] 14π(2-y)(1-y^2) dy = 112π/3

Using discs (washers) of thickness dx, we have
v = ∫[0,7] π(R^2-r^2) dx
where R = 2+√(x/7) and r = 2-√(x/7)
v = ∫[0,7] π((2+√(x/7))^2-(2-√(x/7))^2) dx = 112π/3