Answer
$V = \int_1^2 {2\pi x\ln x} dx$
Work Step by Step
$$\eqalign{
& y = \ln x,{\text{ }}y = 0,{\text{ }}x = 2,{\text{ about the }}y{\text{ - axis}} \cr
& {\text{Find the intersection points with the }}x{\text{ - axis}} \cr
& {\text{Let }}y = 0 \cr
& \ln x = 0 \cr
& x = 1 \cr
& {\text{From the graph shown below}}{\text{, we can see that is easier apply}} \cr
& {\text{the shell method about the }}y{\text{ - axis}}{\text{.}} \cr
& V = \int_a^b {2\pi x} \left( {f\left( x \right) - g\left( x \right)} \right)dx \cr
& {\text{Therefore}}{\text{,}} \cr
& V = \int_1^2 {2\pi x\left( {\ln x - 0} \right)} dx \cr
& V = \int_1^2 {2\pi x\ln x} dx \cr} $$
