Answer
$\text{The volume is}$
\begin{align}
V =\frac{7\pi}{4}
\end{align}
Work Step by Step
$\text{It is given that}$
\begin{align}
y = \frac{1}{x^3}; \ x = 1; \ x = 2 \ \ and \ \ y = 0
\end{align}
$\text{The given region is revolved about x = -1. Meaning that the cross }$
$\text{section generates a cylindircal surface of height $\frac{1}{x^3}$ and radius (x+1).}$
$\text{Thus, the volume is}$
\begin{align}
V = 2\pi \int_1^2 \frac{1}{x^3}(x+1) \ dx = 2\pi \left[-\frac{1}{x} - \frac{1}{2x^2} \right]_1^2 = 2\pi \times \frac{7}{8} = \frac{7\pi}{4}
\end{align}
