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The diagram shows the curve \(y = \sin x \cos 2 x\), for \(0 \leqslant x \leqslant \pi\), and a maximum point \(M\), where \(x = a\). The shaded region between the curve and the \(x\)-axis is denoted by \(R\).
- Find the value of \(a\) correct to 2 decimal places.
- Find the exact area of the region \(R\), giving your answer in simplified form.