15.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ce06b37a-aa57-4256-bec8-7277c8a9fc65-44_851_1506_212_260}
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\caption{Figure 3}
\end{figure}
A design for a logo consists of two finite regions \(R _ { 1 }\) and \(R _ { 2 }\), shown shaded in Figure 3 .
The region \(R _ { 1 }\) is bounded by the straight line \(l\) and the curve \(C\).
The region \(R _ { 2 }\) is bounded by the straight line \(l\), the curve \(C\) and the line with equation \(x = 5\)
The line \(l\) has equation \(y = 8 x + 38\)
The curve \(C\) has equation \(y = 4 x ^ { 2 } + 6\)
Given that the line \(l\) meets the curve \(C\) at the points \(( - 2,22 )\) and \(( 4,70 )\), use integration to find
- the area of the larger lower region, labelled \(R _ { 1 }\)
- the exact value of the total area of the two shaded regions.
Given that
$$\frac { \text { Area of } R _ { 1 } } { \text { Area of } R _ { 2 } } = k$$
- find the value of \(k\).
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