Multiple Region or Composite Area

15. \begin{figure}[h] \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

1. the area of the larger lower region, labelled \(R _ { 1 }\)
2. the exact value of the total area of the two shaded regions. Given that $$\frac { \text { Area of } R _ { 1 } } { \text { Area of } R _ { 2 } } = k$$
3. find the value of \(k\).

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1. In this question you must show all stages of your working.

\section*{Solutions based entirely on calculator technology are not acceptable.} \begin{figure}[h] \end{figure} Figure 3 shows

• the curve \(C\) with equation \(y = x ^ { 2 } - 4 x + 5\)
• the line \(l\) with equation \(y = 2\)

The curve \(C\) intersects the \(y\)-axis at the point \(D\).

1. Write down the coordinates of \(D\). The curve \(C\) intersects the line \(l\) at the points \(E\) and \(F\), as shown in Figure 3.
2. Find the \(x\) coordinate of \(E\) and the \(x\) coordinate of \(F\). Shown shaded in Figure 3 is
   • the region \(R _ { 1 }\) which is bounded by \(C , l\) and the \(y\)-axis
   • the region \(R _ { 2 }\) which is bounded by \(C\) and the line segments \(E F\) and \(D F\)
   Given that \(\frac { \text { area of } R _ { 1 } } { \text { area of } R _ { 2 } } = k\), where \(k\) is a constant,
3. use algebraic integration to find the exact value of \(k\), giving your answer as a simplified fraction.

7 \includegraphics[max width=\textwidth, alt={}, center]{d6430776-0b87-4e5e-8f78-c6228ee163d5-5_647_741_260_260} The diagram shows part of the curve \(y = ( 5 - x ) ( x - 1 )\) and the line \(x = a\).
Given that the total area of the regions shaded in the diagram is 19 units \({ } ^ { 2 }\), determine the exact value of \(a\).

\includegraphics{figure_2} Figure 2 shows part of the curve C with equation y = f(x), where $$f(x) = x^3 - 6x^2 + 5x.$$ The curve crosses the x-axis at the origin O and at the points A and B.

1. Factorise f(x) completely [3 marks]
2. Write down the x-coordinates of the points A and B. [1 marks]
3. Find the gradient of C at A. [3 marks] The region R is bounded by C and the line OA, and the region S is bounded by C and the line AB.
4. Use integration to find the area of the combined regions R and S, shown shaded in Fig. 2. [7 marks]