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The diagram shows the curve \(y = \sin x \cos ^ { 2 } 2 x\) for \(0 \leqslant x \leqslant \frac { 1 } { 4 } \pi\) and its maximum point \(M\).
- Using the substitution \(u = \cos x\), find by integration the exact area of the shaded region bounded by the curve and the \(x\)-axis.
- Find the \(x\)-coordinate of \(M\). Give your answer correct to 2 decimal places.