2.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e375f6ad-4a76-42a0-b7bf-ae47e5cbdaeb-04_479_855_310_566}
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\caption{Figure 1}
\end{figure}
Figure 1 shows a sketch of the curve with equation \(y = \mathrm { e } ^ { x } \sqrt { \sin x } , 0 \leqslant x \leqslant \pi\).
The finite region \(R\), shown shaded in Figure 1, is bounded by the curve and the \(x\)-axis.
- Complete the table below with the values of \(y\) corresponding to \(x = \frac { \pi } { 4 }\) and \(x = \frac { \pi } { 2 }\), giving your answers to 5 decimal places.
| \(x\) | 0 | \(\frac { \pi } { 4 }\) | \(\frac { \pi } { 2 }\) | \(\frac { 3 \pi } { 4 }\) | \(\pi\) |
| \(y\) | 0 | | | 8.87207 | 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.
The curve \(y = \mathrm { e } ^ { x } \sqrt { \sin x } , 0 \leqslant x \leqslant \pi\), has a maximum turning point at \(Q\), shown in Figure 1.
- Find the \(x\) coordinate of \(Q\).