Question:- Find the volume of the solid of revolution generated by revolving the region bounded by the curve $y=x^2+1$ and the line $y=-x+3$ about the $y$- axis.
The region on the right when revolved is overlaped by the region on left when revolved,So the volume is same as revolving the left region about y axis.
$$V=\pi \int_{1}^{3} (y-1) \mathrm {dy}+\pi \int_{3}^{5} (y-1)-(3-y)^2 \mathrm {dy}=\frac{16\pi}{3}$$
Is the above integral for calculating volume correct?