7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e7043e7a-2c8f-425a-8471-f647828cc297-18_1109_958_214_502}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
Figure 1 shows a sketch of part of the curve \(C\) with equation
$$y = x ^ { 3 } - 6 x ^ { 2 } + 9 x + 5$$
The point \(P ( 4,9 )\) lies on \(C\).
- Show that the normal to \(C\) at the point \(P\) has equation
$$x + 9 y = 85$$
The region \(R\), shown shaded in Figure 1, is bounded by the curve \(C\), the \(y\)-axis and the normal to \(C\) at \(P\).
- Showing all your working, calculate the exact area of \(R\).