6.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0e107b51-2fb3-4ad7-8542-5aa0da13b127-20_978_1292_267_328}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
Figure 1 shows a sketch of part of the curves \(C _ { 1 }\) and \(C _ { 2 }\) with equations
$$\begin{array} { l l }
C _ { 1 } : y = x ^ { 3 } - 6 x + 9 & x \geqslant 0
C _ { 2 } : y = - 2 x ^ { 2 } + 7 x - 1 & x \geqslant 0
\end{array}$$
The curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the points \(A\) and \(B\) as shown in Figure 1 .
The point \(A\) has coordinates (1,4).
Using algebra and showing all steps of your working,
- find the coordinates of the point \(B\).
The finite region \(R\), shown shaded in Figure 1, is bounded by \(C _ { 1 }\) and \(C _ { 2 }\)
- Use algebraic integration to find the exact area of \(R\).