2.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cbfbb690-bc85-46e5-a97f-35df4b6f1c84-03_712_1091_248_470}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
Figure 1 shows a sketch of part of the curve with equation \(y = x ^ { 2 } \ln x , x \geqslant 1\)
The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis and the line \(x = 2\)
The table below shows corresponding values of \(x\) and \(y\) for \(y = x ^ { 2 } \ln x\)
| \(x\) | 1 | 1.2 | 1.4 | 1.6 | 1.8 | 2 |
| \(y\) | 0 | 0.2625 | | 1.2032 | 1.9044 | 2.7726 |
- Complete the table above, giving the missing value of \(y\) to 4 decimal places.
- Use the trapezium rule with all the values of \(y\) in the completed table to obtain an estimate for the area of \(R\), giving your answer to 3 decimal places.
- Use integration to find the exact value for the area of \(R\).