6.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2a6d0dba-d948-4124-9740-a88c17b0be65-16_618_1018_228_456}
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\caption{Figure 1}
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Figure 1 shows a sketch of the curve \(C\) with equation \(y = 2 \mathrm { e } ^ { - x } \sqrt { \sin x } , 0 \leqslant x \leqslant \pi\). The finite region \(R\), shown shaded in Figure 1, is bounded by the curve and the \(x\)-axis.
- Complete the table below with the value of \(y\) corresponding to \(x = \frac { \pi } { 2 }\), giving your answer to 5 decimal places.
| \(x\) | 0 | \(\frac { \pi } { 4 }\) | \(\frac { \pi } { 2 }\) | \(\frac { 3 \pi } { 4 }\) | \(\pi\) |
| \(y\) | 0 | 0.76679 | | 0.15940 | 0 |
- Use the trapezium rule, with all the values of \(y\) in the completed table, to obtain an estimate for the area of the region \(R\). Give your answer to 4 decimal places.
- Given \(y = 2 \mathrm { e } ^ { - x } \sqrt { \sin x }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) for \(0 < x < \pi\).
The curve \(C\) has a maximum turning point when \(x = a\).
- Use your answer to part (c) to find the value of \(a\), giving your answer to 3 decimal places.