Question

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

Answer 1

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

Answer 2

\[ y = \sqrt{x} \]
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 \( -\pi y \ dy \)

Answer 3

that's all there is to it?

Answer 4

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