5
\includegraphics[max width=\textwidth, alt={}, center]{4c71f68a-efb9-4408-bf03-874e0d4426d5-2_458_807_1786_667}
The diagram shows the curve
$$y = 8 \sin \frac { 1 } { 2 } x - \tan \frac { 1 } { 2 } x$$
for \(0 \leqslant x < \pi\). The \(x\)-coordinate of the maximum point is \(\alpha\) and the shaded region is enclosed by the curve and the lines \(x = \alpha\) and \(y = 0\).
- Show that \(\alpha = \frac { 2 } { 3 } \pi\).
- Find the exact value of the area of the shaded region.