Standard Trapezium Rule Application

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\includegraphics[max width=\textwidth, alt={}, center]{0355f624-3a35-4b9e-8520-af011a0fb6db-2_499_787_922_678} The diagram shows the part of the curve \(y = \sqrt { } ( 2 - \sin x )\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\).

1. Use the trapezium rule with 2 intervals to estimate the value of $$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sqrt { } ( 2 - \sin x ) \mathrm { d } x$$ giving your answer correct to 2 decimal places.
2. The line \(y = x\) intersects the curve \(y = \sqrt { } ( 2 - \sin x )\) at the point \(P\). Use the iterative formula $$x _ { n + 1 } = \sqrt { } \left( 2 - \sin x _ { n } \right)$$ to determine the \(x\)-coordinate of \(P\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

1. (a) Given that

$$y = \ln \left( 3 + x ^ { 2 } \right)$$ complete the table with the value of \(y\) corresponding to \(x = 3\), giving your answer to 4 significant figures. In part (b) you must show all stages of your working. \section*{Solutions relying entirely on calculator technology are not acceptable.} (b) Use Simpson's rule with all the values of \(y\) in the completed table to estimate, to 3 significant figures, the value of $$\int _ { 2 } ^ { 5 } \ln \left( 3 + x ^ { 2 } \right) \mathrm { d } x$$ (c) Using your answer to part (b) and making your method clear, estimate the value of $$\int _ { 2 } ^ { 5 } \ln \sqrt { \left( 3 + x ^ { 2 } \right) } \mathrm { d } x$$