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The diagram shows the curve \(y = \sin 2 x \cos ^ { 2 } x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\), and its maximum point \(M\).
- Using the substitution \(u = \sin x\), find the exact area of the region bounded by the curve and the \(x\)-axis.
- Find the exact \(x\)-coordinate of \(M\).
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