Answer
$\pi$
Work Step by Step
We apply the disk method (see graph below)
$V=\displaystyle \int_{a}^{b}\pi[R(x)]^{2}dx$
The solid is symmetric, so we take bounds from 0 to $\ln\sqrt{3}$
$ y=2\displaystyle \pi\int_{0}^{\ln\sqrt{3}}{\rm sech}^{2}xdx=\quad$
See the table "Integral formulas for hyperbolic functions"
$=2\pi[\tanh x]_{0}^{\ln\sqrt{3}}$
Evaluate using $\displaystyle \tanh x=\frac{e^{x}-e^{-x}}{e^{x}+e^{x}}\qquad\left[\begin{array}{l}
\tanh 0=\frac{1-1}{1+1}=0\\
\\
\tanh(\ln\sqrt{3})=\frac{\sqrt{3}-\frac{1}{\sqrt{3}}}{\sqrt{3}+\frac{1}{\sqrt{3}}}\cdot\frac{\sqrt{3}}{\sqrt{3}}=\frac{3-1}{3+1}=\frac{1}{2}
\end{array}\right]$
$=2\displaystyle \pi[\frac{1}{2}-0]$
$=\pi$
