9709 P33 - Jun 2012 - Q7
1850
The diagram shows part of the curve \(y = \\cos(\sqrt{x})\) for \(x \geq 0\), where \(x\) is in radians. The shaded region between the curve, the axes and the line \(x = p^2\), where \(p > 0\), is denoted by \(R\). The area of \(R\) is equal to 1.
(i) Use the substitution \(x = u^2\) to find \(\int_0^{p^2} \cos(\sqrt{x}) \, dx\). Hence show that \(\sin p = \frac{3 - 2 \cos p}{2p}\).
(ii) Use the iterative formula \(p_{n+1} = \sin^{-1} \left( \frac{3 - 2 \cos p_n}{2p_n} \right)\), with initial value \(p_1 = 1\), to find the value of \(p\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
