8.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{571780c2-945b-4636-b7c3-0bd558d28710-10_611_831_210_575}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
Figure 2 shows a sketch of part of the curve \(C\) with equation
$$y = x ^ { 3 } - 10 x ^ { 2 } + k x$$
where \(k\) is a constant.
The point \(P\) on \(C\) is the maximum turning point.
Given that the \(x\)-coordinate of \(P\) is 2 ,
- show that \(k = 28\).
The line through \(P\) parallel to the \(x\)-axis cuts the \(y\)-axis at the point \(N\). The region \(R\) is bounded by \(C\), the \(y\)-axis and \(P N\), as shown shaded in Figure 2.
- Use calculus to find the exact area of \(R\).