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The diagram shows the curve \(y = x ^ { 2 } \cos 2 x\) for \(0 \leqslant x \leqslant \frac { 1 } { 4 } \pi\). The curve has a maximum point at \(M\) where \(x = p\).
- Show that \(p\) satisfies the equation \(p = \frac { 1 } { 2 } \tan ^ { - 1 } \left( \frac { 1 } { p } \right)\).
- Use the iterative formula \(p _ { n + 1 } = \frac { 1 } { 2 } \tan ^ { - 1 } \left( \frac { 1 } { p _ { n } } \right)\) to determine the value of \(p\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
- Find, showing all necessary working, the exact area of the region bounded by the curve and the \(x\)-axis.