10.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{752efc6c-8d0e-46a6-b75d-5125956969d8-28_761_1120_258_411}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
Figure 2 shows a sketch of part of the curve with equation
$$y = 4 x ^ { 3 } + 9 x ^ { 2 } - 30 x - 8 , \quad - 0.5 \leqslant x \leqslant 2.2$$
The curve has a turning point at the point \(A\).
- Using calculus, show that the \(x\) coordinate of \(A\) is 1
The curve crosses the \(x\)-axis at the points \(B ( 2,0 )\) and \(C \left( - \frac { 1 } { 4 } , 0 \right)\)
The finite region \(R\), shown shaded in Figure 2, is bounded by the curve, the line \(A B\), and the \(x\)-axis.
- Use integration to find the area of the finite region \(R\), giving your answer to 2 decimal places.