14.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f7994129-07ee-4f6d-9531-08a15a38b794-26_567_412_212_824}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{figure}
Figure 4 shows a sketch of part of the curve \(C\) with equation
$$y = \frac { x ^ { 2 } \ln x } { 3 } - 2 x + 5 , \quad x > 0$$
The finite region \(S\), shown shaded in Figure 4, is bounded by the curve \(C\), the line with equation \(x = 1\), the \(x\)-axis and the line with equation \(x = 3\)
The table below shows corresponding values of \(x\) and \(y\) with the values of \(y\) given to 4 decimal places as appropriate.
| \(x\) | 1 | 1.5 | 2 | 2.5 | 3 |
| \(y\) | 3 | 2.3041 | 1.9242 | 1.9089 | 2.2958 |
- Use the trapezium rule, with all the values of \(y\) in the table, to obtain an estimate for the area of \(S\), giving your answer to 3 decimal places.
- Explain how the trapezium rule could be used to obtain a more accurate estimate for the area of \(S\).
- Show that the exact area of \(S\) can be written in the form \(\frac { a } { b } + \ln c\), where \(a , b\) and \(c\) are integers to be found.
(In part c, solutions based entirely on graphical or numerical methods are not acceptable.)