Answer
\[V=2{{\pi }^{2}}+16\pi \]
Work Step by Step
\[\begin{align}
& y=2\sin x,\text{ the }x\text{-axis on the interval }\left[ 0,\pi \right],\text{ revolved about} \\
& \text{the line }y=-2 \\
& \text{Apply the Washer Method about the }x\text{-axis} \\
& V=\int_{a}^{b}{\pi \left[ f{{\left( x \right)}^{2}}-g{{\left( x \right)}^{2}} \right]}dx \\
& \text{The axis of rotation is about the line }y=-2,\text{ then} \\
& V=\int_{a}^{b}{\pi \left[ {{\left( f\left( x \right)+2 \right)}^{2}}-{{\left( g\left( x \right)+2 \right)}^{2}} \right]}dx \\
& \text{Substitute }f\left( x \right)=2\sin x\text{ and }g\left( x \right)=0,\text{ for }\left[ \underbrace{0}_{a},\underbrace{\pi }_{b} \right] \\
& V=\int_{0}^{\pi }{\pi \left[ {{\left( 2\sin x+2 \right)}^{2}}-{{\left( 0+2 \right)}^{2}} \right]}dx \\
& V=\int_{0}^{\pi }{\pi \left[ {{\left( 2\sin x+2 \right)}^{2}}-4 \right]}dx \\
& \text{Simplify the integrand} \\
& V=\pi \int_{0}^{\pi }{\left( 4{{\sin }^{2}}x+8\sin x+4-4 \right)}dx \\
& V=\pi \int_{0}^{\pi }{\left( 4{{\sin }^{2}}x+8\sin x \right)}dx \\
& V=4\pi \int_{0}^{\pi }{\left( {{\sin }^{2}}x+2\sin x \right)}dx \\
& \text{Use the identity }{{\sin }^{2}}x=\frac{1}{2}-\frac{\cos 2x}{2} \\
& V=4\pi \int_{0}^{\pi }{\left( \frac{1}{2}-\frac{\cos 2x}{2}+2\sin x \right)}dx \\
& \text{Integrate} \\
& V=4\pi \left[ \frac{1}{2}x-\frac{\sin 2x}{4}-2\cos x \right]_{0}^{\pi } \\
& \text{Evaluate} \\
& V=4\pi \left[ \frac{1}{2}\pi -\frac{\sin 2\pi }{4}-2\cos \pi \right]-4\pi \left[ \frac{1}{2}0-\frac{\sin 20}{4}-2\cos 0 \right] \\
& V=4\pi \left[ \frac{1}{2}\pi +2 \right]-4\pi \left[ -2 \right] \\
& V=2{{\pi }^{2}}+8\pi +8\pi \\
& V=2{{\pi }^{2}}+16\pi \\
\end{align}\]
