Answer
$\displaystyle{V=9\pi\ln3-4\pi}\\$
Work Step by Step
$\displaystyle{y=e^{x}}\\
\displaystyle{\ln y=\ln e^{x}}\\
\displaystyle{\ln y=x\ln e}\\
\displaystyle{x=\ln y}$
$\displaystyle{V=\int_1^3 (2\pi y)\left(\ln y\right)\ dy}\\
\displaystyle{V=2\pi\int_1^3 y\ln y\ dx}\\$
$\displaystyle \left[\begin{array}{ll} u=\ln y & dv=y \\ & \\ du=\frac{1}{y} & v=\frac{1}{2}y^2 \end{array}\right]$ Integration by parts
$\displaystyle{V=2\pi\left[\frac{1}{2}y^2\ln y\right]_1^3-2\pi\int_1^3\frac{1}{2}y^2\times\frac{1}{y}\ dy}\\
\displaystyle{V=2\pi\left[\frac{9}{2}\ln3\right]-\pi\int_1^3y\ dy}\\
\displaystyle{V=9\pi\ln3-\pi\left[\frac{1}{2}y^2\right]_1^3}\\
\displaystyle{V=9\pi\ln3-4\pi}\\$
