Answer
$V = 6\pi $
Work Step by Step
$$\eqalign{
& y = \frac{1}{x}{\text{, }}y = 0,{\text{ }}x = 1,{\text{ }}x = 4 \cr
& {\text{Apply the shell method about the }}y{\text{ - axis}}{\text{.}} \cr
& V = \int_a^b {2\pi x} f\left( x \right)dx,{\text{ where }}0 \leqslant a \leqslant b \cr
& {\text{Therefore}}{\text{,}} \cr
& V = \int_1^4 {2\pi x\left( {\frac{1}{x}} \right)} dx \cr
& V = 2\pi \int_1^4 {dx} \cr
& {\text{Integrating}} \cr
& V = 2\pi \left[ x \right]_1^4 \cr
& {\text{Evaluating}} \cr
& V = 2\pi \left[ {4 - 1} \right] \cr
& V = 6\pi \cr
& \cr} $$