2 Fig. 8 shows the curve \(y = 3 \ln x + x - x ^ { 2 }\).
The curve crosses the \(x\)-axis at P and Q , and has a turning point at R . The \(x\)-coordinate of Q is approximately 2.05 .
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{72893fd5-bc8e-433b-8358-f7979b2da636-2_717_830_606_693}
\captionsetup{labelformat=empty}
\caption{Fig. 8}
\end{figure}
- Verify that the coordinates of P are \(( 1,0 )\).
- Find the coordinates of R , giving the \(y\)-coordinate correct to 3 significant figures.
Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\), and use this to verify that R is a maximum point.
- Find \(\int \ln x \mathrm {~d} x\).
Hence calculate the area of the region enclosed by the curve and the \(x\)-axis between P and Q , giving your answer to 2 significant figures.