5.
\begin{figure}[h]
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\caption{Figure 1}
\includegraphics[alt={},max width=\textwidth]{cb12f63c-f4d0-4eb8-b4a5-0ad12f926b1a-3_668_1172_1231_354}
\end{figure}
Figure 1 shows a graph of \(y = x \sqrt { } \sin x , 0 < x < \pi\). The maximum point on the curve is \(A\).
- Show that the \(x\)-coordinate of the point \(A\) satisfies the equation \(2 \tan x + x = 0\).
The finite region enclosed by the curve and the \(x\)-axis is shaded as shown in Fig. 1.
A solid body \(S\) is generated by rotating this region through \(2 \pi\) radians about the \(x\)-axis. - Find the exact value of the volume of \(S\).
(7)