13.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4c08fbab-283e-4c92-89a4-10f68f37e133-22_536_929_223_504}
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\caption{Figure 5}
\end{figure}
Figure 5 shows a sketch of part of the curve with equation \(y = 2 - \ln x , x > 0\)
The finite region \(R\), shown shaded in Figure 5, is bounded by the curve, the \(x\)-axis and the line with equation \(x = \mathrm { e }\).
The table below shows corresponding values of \(x\) and \(y\) for \(y = 2 - \ln x\)
| \(x\) | e | \(\frac { \mathrm { e } + \mathrm { e } ^ { 2 } } { 2 }\) | \(\mathrm { e } ^ { 2 }\) |
| \(y\) | 1 | | 0 |
- Complete the table giving the 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 by parts to show that \(\int ( \ln x ) ^ { 2 } \mathrm {~d} x = x ( \ln x ) ^ { 2 } - 2 x \ln x + 2 x + c\)
The area \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
- Use calculus to find the exact volume of the solid generated.
Write your answer in the form \(\pi \mathrm { e } ( p \mathrm { e } + q )\), where \(p\) and \(q\) are integers to be found.