Curve-Line-Axis Bounded Region

12 \includegraphics[max width=\textwidth, alt={}, center]{fdd6e942-b5bc-4369-8587-6de120459776-18_557_677_264_733} The diagram shows a curve with equation \(y = 4 x ^ { \frac { 1 } { 2 } } - 2 x\) for \(x \geqslant 0\), and a straight line with equation \(y = 3 - x\). The curve crosses the \(x\)-axis at \(A ( 4,0 )\) and crosses the straight line at \(B\) and \(C\).

1. Find, by calculation, the \(x\)-coordinates of \(B\) and \(C\).
2. Show that \(B\) is a stationary point on the curve.
3. Find the area of the shaded region.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

14. \begin{figure}[h] \end{figure} Diagram NOT drawn to scale Figure 2 shows part of the line \(l\) with equation \(y = 2 x - 3\) and part of the curve \(C\) with equation \(y = x ^ { 2 } - 2 x - 15\) The line \(l\) and the curve \(C\) intersect at the points \(A\) and \(B\) as shown.

1. Use algebra to find the coordinates of \(A\) and the coordinates of \(B\). In Figure 2, the shaded region \(R\) is bounded by the line \(l\), the curve \(C\) and the positive \(x\)-axis.
2. Use integration to calculate an exact value for the area of \(R\).

16. \begin{figure}[h] \end{figure} The straight line \(l\) with equation \(y = \frac { 1 } { 2 } x + 1\) cuts the curve \(C\), with equation \(y = x ^ { 2 } - 4 x + 3\), at the points \(P\) and \(Q\), as shown in Figure 2

1. Use algebra to find the coordinates of the points \(P\) and \(Q\). The curve \(C\) crosses the \(x\)-axis at the points \(T\) and \(S\).
2. Write down the coordinates of the points \(T\) and \(S\). The finite region \(R\) is shown shaded in Figure 2. This region \(R\) is bounded by the line segment \(P Q\), the line segment \(T S\), and the \(\operatorname { arcs } P T\) and \(S Q\) of the curve.
3. Use integration to find the exact area of the shaded region \(R\).

14. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of part of the line \(l\) with equation \(y = 8 - x\) and part of the curve \(C\) with equation \(y = 14 + 3 x - 2 x ^ { 2 }\) The line \(l\) and the curve \(C\) intersect at the point \(A\) and the point \(B\) as shown.

1. Use algebra to find the coordinates of \(A\) and the coordinates of \(B\). The region \(R\), shown shaded in Figure 5, is bounded by the coordinate axes, the line \(l\), and the curve \(C\).
2. Use algebraic integration to calculate the exact area of \(R\).

15. \begin{figure}[h] \end{figure} The straight line \(l\) with equation \(y = 5 - 3 x\) cuts the curve \(C\), with equation \(y = 20 x - 12 x ^ { 2 }\), at the points \(P\) and \(Q\), as shown in Figure 3.

1. Use algebra to find the exact coordinates of the points \(P\) and \(Q\). The finite region \(R\), shown shaded in Figure 3, is bounded by the line \(l\), the \(x\)-axis and the curve \(C\).
2. Use calculus to find the exact area of \(R\).

16. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the curve with equation \(y = 2 x ^ { 2 } - 11 x + 12\). The curve crosses the \(y\)-axis at the point \(A\) and crosses the \(x\)-axis at the points \(B\) and \(C\).

1. Find the coordinates of the points \(A , B\) and \(C\). The point \(D\) lies on the curve such that the line \(A D\) is parallel to the \(x\)-axis. The finite region \(R\), shown shaded in Figure 4, is bounded by the curve, the line \(A C\) and the line \(A D\).
2. Use algebraic integration to find the exact area of \(R\).

10. \begin{figure}[h] \end{figure} The finite region \(R\), which is shown shaded in Figure 1, is bounded by the coordinate axes, the straight line \(l\) with equation \(y = \frac { 1 } { 3 } x + 5\) and the curve \(C\) with equation \(y = 4 x ^ { \frac { 1 } { 2 } } - x + 5 , x \geqslant 0\) The line \(l\) meets the curve \(C\) at the point \(D\) on the \(y\)-axis and at the point \(E\), as shown in Figure 1.

1. Use algebra to find the coordinates of the points \(D\) and \(E\). The curve \(C\) crosses the \(x\)-axis at the point \(F\).
2. Verify that the \(x\) coordinate of \(F\) is 25
3. Use algebraic integration to find the exact area of the shaded region \(R\).

7. \begin{figure}[h] \end{figure} The curve \(C\) has equation \(y = x ^ { 2 } - 5 x + 4\). It cuts the \(x\)-axis at the points \(L\) and \(M\) as shown in Figure 2.

1. Find the coordinates of the point \(L\) and the point \(M\).
2. Show that the point \(N ( 5,4 )\) lies on \(C\).
3. Find \(\int \left( x ^ { 2 } - 5 x + 4 \right) \mathrm { d } x\). The finite region \(R\) is bounded by \(L N , L M\) and the curve \(C\) as shown in Figure 2.
4. Use your answer to part (c) to find the exact value of the area of \(R\).
   \section*{LU}

9 The diagram shows the curve \(\mathrm { y } = 3 - \sqrt { \mathrm { x } }\). \includegraphics[max width=\textwidth, alt={}, center]{a0d9573f-8273-4562-a2d3-07f15d9da1af-6_810_1008_1155_283}

1. Draw the line \(\mathrm { y } = 5 \mathrm { x } - 1\) on the copy of the diagram in the Printed Answer Booklet.
2. In this question you must show detailed reasoning. Determine the exact area of the region bounded by the curve \(y = 3 - \sqrt { x }\), the lines \(y = 5 x - 1\) and \(x = 4\) and the \(x\)-axis.

\includegraphics{figure_14} The diagram above shows the curve with equation $$y = (x-4)^2, \quad x \in \mathbb{R},$$ intersected by the straight line with equation \(y = 4\), at the points \(A\) and \(B\). The curve meets the \(y\) axis at the point \(C\). Calculate the exact area of the shaded region, bounded by the curve and the straight line segments \(AB\) and \(BC\). [8]