Answer
$$V = 2{\pi ^2}$$
Work Step by Step
$$\eqalign{
& {\text{Using the Shell method, the Volume of the solid can be}} \cr
& {\text{calculated by:}} \cr
& V\int_a^b {2\pi x\left( {f\left( x \right) - g\left( x \right)} \right)dx} \cr
& {\text{With }}a = 0,\,\,\,\,b = \pi ,\,\,\,\,f\left( x \right) = \sin x{\text{ and }}g\left( x \right) = 0 \cr
& {\text{Then,}} \cr
& V = \int_0^\pi {2\pi x\left( {\sin x - 0} \right)dx} \cr
& V = 2\pi \int_0^\pi {x\sin xdx} \cr
& {\text{Integrating by parts}} \cr
& \,\,\,u = x,\,\,\,\,dv = \sin x \cr
& \,\,du = dx,\,\,\,\,\,v = - \cos x \cr
& V = 2\pi \left[ { - x\cos x} \right]_0^\pi + 2\pi \int_0^\pi {\cos xdx} \cr
& V = 2\pi \left( { - \pi \cos \pi } \right) + 2\pi \left[ {\sin x} \right]_0^\pi \cr
& V = 2\pi \left( { - \pi \left( { - 1} \right)} \right) + 0 \cr
& V = 2{\pi ^2} \cr} $$
