11.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{08ede5ea-85e9-44eb-be6a-5878096734e2-34_705_837_248_614}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
Figure 2 shows a sketch of part of the curve with equation
$$y = ( \ln x ) ^ { 2 } \quad x > 0$$
The finite region \(R\), shown shaded in Figure 2, is bounded by the curve, the line with equation \(x = 2\), the \(x\)-axis and the line with equation \(x = 4\)
The table below shows corresponding values of \(x\) and \(y\), with the values of \(y\) given to 4 decimal places.
| \(x\) | 2 | 2.5 | 3 | 3.5 | 4 |
| \(y\) | 0.4805 | 0.8396 | 1.2069 | 1.5694 | 1.9218 |
- Use the trapezium rule, with all the values of \(y\) in the table, to obtain an estimate for the area of \(R\), giving your answer to 3 significant figures.
- Use algebraic integration to find the exact area of \(R\), giving your answer in the form
$$y = a ( \ln 2 ) ^ { 2 } + b \ln 2 + c$$
where \(a\), \(b\) and \(c\) are integers to be found.