3 Fig. 3 shows the curve \(y = \ln x\) and part of the line \(y = 2\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9296c786-a42a-4aa5-b326-39adbb544cbc-02_250_550_979_753}
\captionsetup{labelformat=empty}
\caption{Fig. 3}
\end{figure}
The shaded region is rotated through \(360 ^ { \circ }\) about the \(y\)-axis.
- Show that the volume of the solid of revolution formed is given by \(\int _ { 0 } ^ { 2 } \pi \mathrm { e } ^ { 2 y } \mathrm {~d} y\).
- Evaluate this, leaving your answer in an exact form.