What is the volume of the flying saucer? > What is the volume of the flying saucer that comes from rotating $y=\sin x\;(0\le x\le\pi)$ around the $y$ axis? Here is a diagram for visualization aid: ^2\,dy\\\ &=\pi\left(y\left(\sin^{-1}y\right)^2+2\sin^{-1}y\sqrt{1-y^2}-2y\right)\biggr|_0^1\\\ &=\frac{\pi^3}4-2\pi \end{align}$$ However, the answer key says $V=\int_0^\pi2\pi x\sin x\,dx=2\pi^2$. Where was I wrong?

I figured out why the answer could be $\int_0^\pi2\pi x\sin x\,dx=2\pi^2$. The flying saucer can be visualized by looking at the following longitudinal section: ![Longitudinal section of flying saucer]( By using cylindrical shells, we have $V=\int_a^b2\pi xh\,dx$, where $a=0$, $b=\pi$ and $h=\sin x$. This leads us to the answer in the key $\int_0^\pi2\pi x\sin x\,dx=2\pi\cdot\pi=2\pi^2$.