5.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{515f245f-9c5b-4263-ab2c-0a4f96f3bff0-14_547_1084_269_420}
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\caption{Figure 2}
\end{figure}
Figure 2 shows a sketch of part of the graph of the curves \(C _ { 1 }\) and \(C _ { 2 }\)
The curves intersect when \(x = 2.5\) and when \(x = 4\)
A table of values for some points on the curve \(C _ { 1 }\) is shown below, with \(y\) values given to 3 decimal places as appropriate.
| \(x\) | 2.5 | 2.75 | 3 | 3.25 | 3.5 | 3.75 | 4 |
| \(y\) | 5.453 | 7.764 | 9.375 | 9.964 | 9.367 | 7.626 | 5 |
Using the trapezium rule with all the values of \(y\) in the table,
- find, to 2 decimal places, an estimate for the area bounded by the curve \(C _ { 1 }\), the line with equation \(x = 2.5\), the \(x\)-axis and the line with equation \(x = 4\)
The curve \(C _ { 2 }\) has equation
$$y = x ^ { \frac { 3 } { 2 } } - 3 x + 9 \quad x > 0$$
- Find \(\int \left( x ^ { \frac { 3 } { 2 } } - 3 x + 9 \right) \mathrm { d } x\)
The region \(R\), shown shaded in Figure 2, is bounded by the curves \(C _ { 1 }\) and \(C _ { 2 }\)
- Use the answers to part (a) and part (b) to find, to one decimal place, an estimate for the area of the region \(R\).
(3)