Answer
\begin{aligned}
(a)V &=\int_{2\pi}^{3\pi } 2\pi x\sin(x) d x\\
(b)V&= 98.69604 \end{aligned}
Work Step by Step
Given
$$y=\sin x, y=0, x=2 \pi, x=3 \pi ; \quad \text { about the } y \text {-axis }$$
The volume of the solid when the region rotate about y -axis given by
\begin{aligned}
V&= \int_a^b 2\pi r(x) h(x)dx
\end{aligned}
where $r(x)$ is the radius of shell and $h(x)$ is the height.
Here
\begin{aligned}
r(x)&= x,\\
h(x)&= \sin(x)
\end{aligned}
Hence
\begin{aligned}
V &=\int_{2\pi}^{3\pi } 2\pi x\sin(x) d x\\
&=2 \pi \left( -x\cos(x)+\int_{2\pi}^{3\pi 2} \cos(x)dx \right)\\
&=2 \pi \left( -x\cos(x)+ \sin(x)\right)\bigg|_{2\pi}^{3\pi }\\
&= 10\pi ^2\approx 98.69604
\end{aligned}