The diagram shows the curve

\(y = 8 \sin \frac{1}{2}x - \tan \frac{1}{2}x\)

for \(0 \leq x < \pi\). The \(x\)-coordinate of the maximum point is \(\alpha\) and the shaded region is enclosed by the curve and the lines \(x = \alpha\) and \(y = 0\).

(i) Show that \(\alpha = \frac{2}{3}\pi\).

(ii) Find the exact value of the area of the shaded region.