2.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c2622c33-9436-4254-a728-10ba4703a28c-03_655_1079_207_427}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
Figure 1 shows the finite region \(R\) bounded by the \(x\)-axis, the \(y\)-axis and the curve with equation \(y = 3 \cos \left( \frac { x } { 3 } \right) , 0 \leqslant x \leqslant \frac { 3 \pi } { 2 }\).
The table shows corresponding values of \(x\) and \(y\) for \(y = 3 \cos \left( \frac { x } { 3 } \right)\).
| \(x\) | 0 | \(\frac { 3 \pi } { 8 }\) | \(\frac { 3 \pi } { 4 }\) | \(\frac { 9 \pi } { 8 }\) | \(\frac { 3 \pi } { 2 }\) |
| \(y\) | 3 | 2.77164 | 2.12132 | | 0 |
- Complete the table above giving the missing value of \(y\) to 5 decimal places.
- Using the trapezium rule, with all the values of \(y\) from the completed table, find an approximation for the area of \(R\), giving your answer to 3 decimal places.
- Use integration to find the exact area of \(R\).