12.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{03548211-79cb-4629-b6ca-aa9dfcc77a33-21_615_732_233_605}
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\caption{Figure 4}
\end{figure}
Figure 4 shows a sketch of part of the curve \(C\) with equation
$$y = \frac { x ^ { 2 } \ln x } { 3 } - 2 x + 4 , \quad x > 0$$
The finite region \(S\), shown shaded in Figure 4, is bounded by the curve \(C\), the \(x\)-axis and the lines with equations \(x = 1\) and \(x = 3\)
- Complete the table below with the value of \(y\) corresponding to \(x = 2\). Give your answer to 4 decimal places.
| \(x\) | 1 | 1.5 | 2 | 2.5 | 3 |
| \(y\) | 2 | 1.3041 | | 0.9089 | 1.2958 |
- Use the trapezium rule, with all the values of \(y\) in the completed table, to obtain an estimate for the area of \(S\), giving your answer to 3 decimal places.
- Use calculus to find the exact area of \(S\).
Give your answer in the form \(\frac { a } { b } + \ln c\), where \(a , b\) and \(c\) are integers.
- Hence calculate the percentage error in using your answer to part (b) to estimate the area of \(S\). Give your answer to one decimal place.
- Explain how the trapezium rule could be used to obtain a more accurate estimate for the area of \(S\).