2 Fig. 8 shows the curve \(y = \frac { x } { \sqrt { x - 2 } }\), together with the lines \(y = x\) and \(x = 11\). The curve meets these lines at P and Q respectively. R is the point \(( 11,11 )\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0b4c4935-998c-404f-8fed-9b39b849168e-2_606_729_485_699}
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\caption{Fig. 8}
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- Verify that the \(x\)-coordinate of P is 3 .
- Show that, for the curve, \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x - 4 } { 2 ( x - 2 ) ^ { \frac { 3 } { 2 } } }\).
Hence find the gradient of the curve at P . Use the result to show that the curve is not symmetrical about \(y = x\).
- Using the substitution \(u = x - 2\), show that \(\int _ { 3 } ^ { 11 } \frac { x } { \sqrt { x - 2 } } \mathrm {~d} x = 25 \frac { 1 } { 3 }\).
Hence find the area of the region PQR bounded by the curve and the lines \(y = x\) and \(x = 11\).