Answer
$ 4 \pi$
Work Step by Step
The shell method to compute the volume of a region: the volume of a solid obtained by rotating the region under $y=f(x)$ over an interval $[m,n]$ about the y-axis is given by:
$V=2 \pi \int_{m}^{n} (Radius) \times (height \ of \ the \ shell) \ dy=2 \pi \int_{m}^{n} (x) \times f(x) \ dx$
Now, $V=2\pi \int_{1}^{3} (x)(x^{-1}) \ dx\\=2 \pi \int_{1}^{3} (1) \ dx \\=2 \pi [x]_{1}^{3} \\= 4 \pi$
