Consider region enclosed by the curves y=sinx and and y=0 for x between 0 and $\displaystyle \pi$.Find the volume of the solid obtained when this region is revolved about the y-axis.

Attempt to solution:

Use shells method

$\displaystyle r=x-0$

$\displaystyle v=2\pi\int_0^{\pi} xsinx dx$

Use integration by parts

$\displaystyle \int udv=-xcosx+\int cosx dx$

$\displaystyle \int udv=2\pi(-xcosx+sinx)$

How come the answer book is saying l am wrong ? What did l do wrong ?