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\includegraphics[max width=\textwidth, alt={}, center]{77a45360-8e1d-4f4f-9830-075d832a14cf-18_549_933_260_605} The diagram shows the curve \(y = \sqrt { x } \cos x\), for \(0 \leqslant x \leqslant \frac { 3 } { 2 } \pi\), and its minimum point \(M\), where \(x = a\). The shaded region between the curve and the \(x\)-axis is denoted by \(R\).

1. Show that \(a\) satisfies the equation \(\tan a = \frac { 1 } { 2 a }\).
2. The sequence of values given by the iterative formula \(a _ { n + 1 } = \pi + \tan ^ { - 1 } \left( \frac { 1 } { 2 a _ { n } } \right)\), with initial value \(x _ { 1 } = 3\), converges to \(a\). Use this formula to determine \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
3. Find the volume of the solid obtained when the region \(R\) is rotated completely about the \(x\)-axis. Give your answer in terms of \(\pi\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.