Answer
$$V = 4\pi $$
Work Step by Step
$$\eqalign{
& {\text{We have }}y = \frac{1}{x},\,\,\,x = 1,\,\,\,x = 3,\,\,y = 0 \cr
& {\text{The volume of the solid can be calculated using cylindrical shells}} \cr
& V = \int_a^b {2\pi x\left[ {f\left( x \right) - g\left( x \right)} \right]dx} \cr
& {\text{Let }}f\left( x \right) = \frac{1}{x}{\text{ and }}g\left( x \right) = 0 \cr
& V = \int_1^3 {2\pi x\left( {\frac{1}{x} - 0} \right)dx} \cr
& V = 2\pi \int_1^3 {dx} \cr
& {\text{Integrating}} \cr
& V = 2\pi \left[ x \right]_1^3 \cr
& V = 2\pi \left( {3 - 1} \right) \cr
& V = 4\pi \cr} $$
