3.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c150d518-2da6-4f08-90c4-a831a31020f9-05_613_793_278_571}
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\caption{Figure 2}
\end{figure}
The region \(R\) is bounded by the curve with equation \(y = \mathrm { e } ^ { x }\), the line \(x = 1\), the line \(x = 2\) and the \(x\)-axis as shown in Figure 2. A uniform solid \(S\) is formed by rotating \(R\) through \(2 \pi\) about the \(x\)-axis.
- Show that the volume of \(S\) is \(\frac { 1 } { 2 } \pi \left( \mathrm { e } ^ { 4 } - \mathrm { e } ^ { 2 } \right)\).
- Find, to 3 significant figures, the \(x\)-coordinate of the centre of mass of \(S\).