7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{12fbfe89-60fe-4890-9a22-2b1988d05d33-11_754_1177_217_388}
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\caption{Figure 3}
\end{figure}
Figure 3 shows a sketch of part of the curve with equation \(y = x ^ { \frac { 1 } { 2 } } \ln 2 x\).
The finite region \(R\), shown shaded in Figure 3, is bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 4\)
- Use the trapezium rule, with 3 strips of equal width, to find an estimate for the area of \(R\), giving your answer to 2 decimal places.
- Find \(\int x ^ { \frac { 1 } { 2 } } \ln 2 x \mathrm {~d} x\).
- Hence find the exact area of \(R\), giving your answer in the form \(a \ln 2 + b\), where \(a\) and \(b\) are exact constants.