2 Fig. 7 shows part of the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = x \sqrt { 1 + x }\). The curve meets the \(x\)-axis at the origin and at the point P .
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9e68f5e0-3394-4962-acd9-25bb31f09f2b-2_487_875_487_624}
\captionsetup{labelformat=empty}
\caption{Fig. 7}
\end{figure}
- Verify that the point P has coordinates \(( - 1,0 )\). Hence state the domain of the function \(\mathrm { f } ( x )\).
- Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 + 3 x } { 2 \sqrt { 1 + x } }\).
- Find the exact coordinates of the turning point of the curve. Hence write down the range of the function.
- Use the substitution \(u = 1 + x\) to show that
$$\int _ { - 1 } ^ { 0 } x \sqrt { 1 + x } \mathrm {~d} x = \int _ { 0 } ^ { 1 } \left( \begin{array} { l l }
u ^ { \frac { 3 } { 2 } } & u ^ { \frac { 1 } { 2 } }
\end{array} \right) \mathrm { d } u .$$
Hence find the area of the region enclosed by the curve and the \(x\)-axis.