Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.

Question

y^2 = 2x, x = 2y; about the y-axis

Answer 1

The curves intersect at (8,4), so using discs (washers) of thickness dy,
v = ∫ π(R^2-r^2) dy
where R=2y and r = y^2/2
v = ∫[0,4] π((2y)^2 - (y^2/2)^2) dy = 512π/15
Or, using shells of thickness dx,
v = ∫ 2πrh dx
where r=x and h = √(2x) - x/2
v = ∫[0,8] 2πx(√(2x) - x/2) dx = 512π/15