Question

find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. sketch the region and a typical disk or washer.
y^2=x, x=2y; about the y-axis


i am confused because i do not know how to set it up because one is Y and the other X i know the formula

Answer 1

The curves intersect at (0,0) and (4,2)
Imagine the washers around the y-axis.

We have

v = ∫[0,2] π(R^2-r^2) dy
where r = y^2 and R = 2y
v = π∫[0,2] (2x)^2 - (y^2)^2 dy
= 64/15 π

Just to check, using shells, we have

v = ∫[0,4] 2πrh dx
where r=x and h = √x - x/2
v = 2π∫[0,4] x(√x - x/2) dx
= 64/15 π