4 Fig. 4 shows a sketch of the region enclosed by the curve \(\sqrt { 1 + \mathrm { e } ^ { - 2 x } }\), the \(x\)-axis, the \(y\)-axis and the line \(x = 1\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{88eadcf5-4335-4016-baf5-d3f74513bbb8-02_517_755_1576_649}
\captionsetup{labelformat=empty}
\caption{Fig. 4}
\end{figure}
Find the volume of the solid generated when this region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Give your answer in an exact form.