5.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e30f0c28-1695-40a1-8e9a-6ea7e29042bf-08_579_1038_258_452}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
Figure 1 shows a sketch of part of the curve \(C\) with equation
$$y = x \cos x , \quad x \in \mathbb { R }$$
The finite region \(R\), shown shaded in Figure 1, is bounded by the curve \(C\) and the \(x\)-axis for \(\frac { 3 \pi } { 2 } \leqslant x \leqslant \frac { 5 \pi } { 2 }\)
- Complete the table below with the exact value of \(y\) corresponding to \(x = \frac { 7 \pi } { 4 }\) and with the exact value of \(y\) corresponding to \(x = \frac { 9 \pi } { 4 }\)
| \(x\) | \(\frac { 3 \pi } { 2 }\) | \(\frac { 7 \pi } { 4 }\) | \(2 \pi\) | \(\frac { 9 \pi } { 4 }\) | \(\frac { 5 \pi } { 2 }\) |
| \(y\) | 0 | | \(2 \pi\) | | 0 |
- Use the trapezium rule, with all five \(y\) values in the completed table, to find an approximate value for the area of \(R\), giving your answer to 4 significant figures.
- Find
$$\int x \cos x d x$$
- Using your answer from part (c), find the exact area of the region \(R\).