Let R be a region delimited by x-axis, y=sqroot(x) and y=2-x. Rotate R around y-axis. Calculate volume. WHY IS MY SETUP WRONG? :S

Question

anonymous · Answer

$$\int\limits_{0}^{1} \pi ((2-y)^2-y)dy$$

phi · Answer

$$ y = \sqrt{x} $$
means
$$ x= y^2 $$
you want x to be the radius of a disc, with area $ \pi x^2 $ and volume $ \pi x^2 dy$ 
$$x^2 = y^4 $$
so you want  $ -\pi y^4 dy$
not just $ -\pi y \ dy $

anonymous · Answer

that's all there is to it?

phi · Answer

I think so. It should be
$$ \int\limits_{0}^{1} \pi ((2-y)^2-y^4) \ dy $$