Question:- Find the volume of the solid generated by revolving the shaded region about $x$-axis( The curve is $x=2y-y^3$).
I think we will not get overlapping volumes due to symmetry of the shaded regions about $x$-axis.
By Shell's method
$$V=2\int_{0}^{\sqrt{2}}2\pi(y)(2y-y^3)\mathrm{dy}$$
Can somebody please verify?