14.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{53865e15-3838-4551-b507-fe49549b87db-40_456_689_269_623}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{figure}
Figure 3 shows a sketch of the curve \(C\) with equation \(y = - x ^ { 2 } + 6 x - 8\). The normal to \(C\) at the point \(P ( 5 , - 3 )\) is the line \(l\), which is also shown in Figure 3.
- Find an equation for \(l\), giving your answer in the form \(a x + b y + c = 0\), where \(a\), b and \(c\) are integers.
The finite region \(R\), shown shaded in Figure 3, is bounded below by the line \(l\) and the curve \(C\), and is bounded above by the \(x\)-axis.
- Find the exact value of the area of \(R\).
(Solutions based entirely on graphical or numerical methods are not acceptable.)