Find the volume formed by rotating the region enclosed by: y=2sqrt(x) and y=x about the line y=4

Answer 1

The curves intersect at (0,0) and (4,4), so

using discs (washers)

v = ∫[0,4] π(R^2-r^2) dx
where R = 4-x and r = 4-2√x
∫[0,4] π((4-x)^2-(4-2√x)^2) dx = 32π/3

using shells,

v = ∫[0,4] 2πrh dy
where r=4-y and h=y-y^2/4
v = ∫[0,4] 2π(4-y)(y-y^2/4) dy = 32π/3