Combined region areas

11 \includegraphics[max width=\textwidth, alt={}, center]{10b2ec29-adca-4313-ae24-bab8b2d9f8a4-16_505_1166_258_486} The diagram shows the line \(x = \frac { 5 } { 2 }\), part of the curve \(y = \frac { 1 } { 2 } x + \frac { 7 } { 10 } - \frac { 1 } { ( x - 2 ) ^ { \frac { 1 } { 3 } } }\) and the normal to the curve at the point \(A \left( 3 , \frac { 6 } { 5 } \right)\).

1. Find the \(x\)-coordinate of the point where the normal to the curve meets the \(x\)-axis.
2. Find the area of the shaded region, giving your answer correct to 2 decimal places.

11 \includegraphics[max width=\textwidth, alt={}, center]{fe4c3555-5736-48c4-b61a-9f6b9a1ee46e-4_598_789_255_678} The diagram shows the curve \(y = \sqrt { } ( 1 + 4 x )\), which intersects the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\). The normal to the curve at \(B\) meets the \(x\)-axis at \(C\). Find

1. the equation of \(B C\),
2. the area of the shaded region.

10 \includegraphics[max width=\textwidth, alt={}, center]{a10ad459-6f86-4845-ba28-e4e394bf3d1e-4_595_800_1548_669} The diagram shows the points \(A ( 1,2 )\) and \(B ( 4,4 )\) on the curve \(y = 2 \sqrt { } x\). The line \(B C\) is the normal to the curve at \(B\), and \(C\) lies on the \(x\)-axis. Lines \(A D\) and \(B E\) are perpendicular to the \(x\)-axis.

1. Find the equation of the normal \(B C\).
2. Find the area of the shaded region.

7. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve \(C\) with equation $$y = x ( x - 1 ) ( x - 5 )$$ Use calculus to find the total area of the finite region, shown shaded in Figure 1, that is between \(x = 0\) and \(x = 2\) and is bounded by \(C\), the \(x\)-axis and the line \(x = 2\).
(9)

6. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve \(C\) with equation $$y = x ( x + 4 ) ( x - 2 )$$ The curve \(C\) crosses the \(x\)-axis at the origin \(O\) and at the points \(A\) and \(B\).

1. Write down the \(x\)-coordinates of the points \(A\) and \(B\). The finite region, shown shaded in Figure 3, is bounded by the curve \(C\) and the \(x\)-axis.
2. Use integration to find the total area of the finite region shown shaded in Figure 3.

7 \includegraphics[max width=\textwidth, alt={}, center]{2ae05b46-6c9f-4aaa-9cba-1116c0ec27d4-3_579_557_858_794} The diagram shows part of the curve \(y = x ^ { 2 } - 3 x\) and the line \(x = 5\).

1. Explain why \(\int _ { 0 } ^ { 5 } \left( x ^ { 2 } - 3 x \right) \mathrm { d } x\) does not give the total area of the regions shaded in the diagram.
2. Use integration to find the exact total area of the shaded regions.

1

1. Find \(\int \left( x ^ { 3 } - 2 x \right) \mathrm { d } x\). The graph below shows part of the curve \(y = x ^ { 3 } - 2 x\) for \(0 \leq x \leq 2\). \includegraphics[max width=\textwidth, alt={}, center]{c55a5f04-3573-4f36-a12c-3755bdd4a45b-2_528_1019_520_321}
2. Show that the area of the shaded region \(P\) is the same as the area of the shaded region \(Q\).

9. \includegraphics[max width=\textwidth, alt={}, center]{e4afa57d-5be3-42a6-ab35-39b0fdcc1681-3_559_732_824_388} The diagram shows the curve with equation \(y = \mathrm { f } ( x )\) which crosses the \(x\)-axis at the origin and at the points \(A\) and \(B\). Given that $$f ^ { \prime } ( x ) = 4 - 6 x - 3 x ^ { 2 }$$

1. find an expression for \(y\) in terms of \(x\),
2. show that \(A\) has coordinates ( \(- 4,0\) ) and find the coordinates of \(B\),
3. find the total area of the two regions bounded by the curve and the \(x\)-axis.

15. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of part of the curve \(C\) with equation $$y = \frac { 32 } { x ^ { 2 } } + 3 x - 8 , \quad x > 0$$ The point \(P ( 4,6 )\) lies on \(C\).
The line \(l\) is the normal to \(C\) at the point \(P\).
The region \(R\), shown shaded in Figure 4, is bounded by the line \(l\), the curve \(C\), the line with equation \(x = 2\) and the \(x\)-axis. Show that the area of \(R\) is 46
(Solutions based entirely on graphical or numerical methods are not acceptable.)

8. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = x ( x + 2 ) ( x - 4 )\).
The region \(R _ { 1 }\) shown shaded in Figure 2 is bounded by the curve and the negative \(x\)-axis.

1. Show that the exact area of \(R _ { 1 }\) is \(\frac { 20 } { 3 }\) The region \(R _ { 2 }\) also shown shaded in Figure 2 is bounded by the curve, the positive \(x\)-axis and the line with equation \(x = b\), where \(b\) is a positive constant and \(0 < b < 4\) Given that the area of \(R _ { 1 }\) is equal to the area of \(R _ { 2 }\)
2. verify that \(b\) satisfies the equation $$( b + 2 ) ^ { 2 } \left( 3 b ^ { 2 } - 20 b + 20 \right) = 0$$ The roots of the equation \(3 b ^ { 2 } - 20 b + 20 = 0\) are 1.225 and 5.442 to 3 decimal places. The value of \(b\) is therefore 1.225 to 3 decimal places.
3. Explain, with the aid of a diagram, the significance of the root 5.442

10. \begin{figure}[h] \end{figure} In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
Figure 3 shows a sketch of part of the curve with equation $$y = 8 x - x ^ { \frac { 5 } { 2 } } \quad x \geqslant 0$$ The curve crosses the \(x\)-axis at the point \(A\).

1. Verify that the \(x\) coordinate of \(A\) is 4 The line \(l _ { 1 }\) is the tangent to the curve at \(A\).
2. Use calculus to show that an equation of line \(l _ { 1 }\) is $$12 x + y = 48$$ The line \(l _ { 2 }\) has equation \(y = 8 x\) The region \(R\), shown shaded in Figure 3, is bounded by the curve, the line \(l _ { 1 }\) and the line \(l _ { 2 }\)
3. Use algebraic integration to find the exact area of \(R\).

5 The function f is defined by \(\mathrm { f } ( x ) = ( x + a ) ( x + 3 a ) ( x - b )\) where \(a\) and \(b\) are positive integers.

1. On the axes in the Printed Answer Booklet, sketch the curve \(y = \mathrm { f } ( x )\).
2. On your sketch show, in terms of \(a\) and \(b\), the coordinates of the points where the curve meets the axes. It is now given that \(a = 1\) and \(b = 4\).
3. Find the total area enclosed between the curve \(y = \mathrm { f } ( x )\) and the \(x\)-axis.

6 \includegraphics[max width=\textwidth, alt={}, center]{a1f4ccbd-f5ed-437a-ae76-c4925ce86e25-05_538_531_264_246} The shape \(A B C\) shown in the diagram is a student's design for the sail of a small boat.
The curve \(A C\) has equation \(y = 2 \log _ { 2 } x\) and the curve \(B C\) has equation \(y = \log _ { 2 } \left( x - \frac { 3 } { 2 } \right) + 3\).

1. State the \(x\)-coordinate of point \(A\).
2. Determine the \(x\)-coordinate of point \(B\).
3. By solving an equation involving logarithms, show that the \(x\)-coordinate of point \(C\) is 2 . It is given that, correct to 3 significant figures, the area of the sail is 0.656 units \(^ { 2 }\).
4. Calculate by how much the area is over-estimated or under-estimated when the curved edges of the sail are modelled as straight lines.

10 In this question you must show detailed reasoning.
The equation of a curve is \(y = 12 x ^ { 3 } - 24 x ^ { 2 } - 60 x + 72\).
Determine the magnitude of the total area bounded by the curve and the \(x\)-axis.

7. \begin{figure}[h] \end{figure} Fig. 2 shows part of the curve \(C\) with equation \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 5 x\).
The curve crosses the \(x\)-axis at the origin \(O\) and at the points \(A\) and \(B\).

1. Factorise \(\mathrm { f } ( x )\) completely.
2. Write down the \(x\)-coordinates of the points \(A\) and \(B\).
3. Find the gradient of \(C\) at \(A\). The region \(R\) is bounded by \(C\) and the line \(O A\), and the region \(S\) is bounded by \(C\) and the line \(A B\).
4. Use integration to find the area of the combined regions \(R\) and \(S\), shown shaded in Fig.2. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{e4d38009-aa70-4c55-9765-c54044ffaa31-4_736_727_338_402}
   \end{figure} Fig. 3 shows a sketch of part of the curve \(C\) with equation \(y = x ^ { 3 } - 7 x ^ { 2 } + 15 x + 3 , x \geq 0\). The point \(P\), on \(C\), has \(x\)-coordinate 1 and the point \(Q\) is the minimum turning point of \(C\).
   1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   2. Find the coordinates of \(Q\).
   3. Show that \(P Q\) is parallel to the \(x\)-axis.
   4. Calculate the area, shown shaded in Fig. 3, bounded by \(C\) and the line \(P Q\).

5

1. Find \(\int \left( x ^ { 3 } - 6 x \right) \mathrm { d } x\).
2. 1. Find \(\int \left( \frac { 4 } { x ^ { 2 } } - 1 \right) \mathrm { d } x\).
   2. The diagram shows part of the curve \(y = \frac { 4 } { x ^ { 2 } } - 1\). \includegraphics[max width=\textwidth, alt={}, center]{35d8bb6d-ff0f-4590-b13d-46e4869e2587-04_707_1283_708_415} The curve crosses the \(x\)-axis at \(( 2,0 )\).
      The shaded region is bounded by the curve, the \(x\)-axis, and the lines \(x = 1\) and \(x = 5\). Calculate the area of the shaded region.

\includegraphics{figure_5} The diagram shows the curves \(y = (1 - 2x)^5\) and \(y = e^{2x-1} - 1\). The curves meet at the point \((\frac{1}{2}, 0)\). Find the exact area of the region (shaded in the diagram) bounded by the \(y\)-axis and by part of each curve. [8]

In this question you must show detailed reasoning. \includegraphics{figure_6} The diagram shows the curves \(y = \sqrt{2x + 9}\) and \(y = 4\mathrm{e}^{-2x} - 1\) which intersect on the \(y\)-axis. The shaded region is bounded by the curves and the \(x\)-axis. Determine the area of the shaded region, giving your answer in the form \(p + q \ln 2\) where \(p\) and \(q\) are constants to be determined. [8]

The graph of \(y = x^3\) is shown. \includegraphics{figure_1} Find the total shaded area. Circle your answer. [1 mark] \(-68\) 60 68 128

The graph of \(y = f(x)\) intersects the \(x\)-axis at \((-3, 0)\), \((0, 0)\) and \((2, 0)\) as shown in the diagram below. \includegraphics{figure_2} The shaded region \(A\) has an area of 189 The shaded region \(B\) has an area of 64 Find the value of \(\int_{-3}^{2} f(x) \, dx\) Circle your answer. [1 mark] \(-253\) \(\quad\) \(-125\) \(\quad\) \(125\) \(\quad\) \(253\)

\includegraphics{figure_5} Figure 5 shows a sketch of part of the curve \(y = 2x + \frac{8}{x^2} - 5\), \(x > 0\). The point \(A(4, \frac{7}{2})\) lies on C. The line \(l\) is the tangent to C at the point A. The region \(R\), shown shaded in figure 5 is bounded by the line \(l\), the curve C, the line with equation \(x = 1\) and the \(x\)-axis. Find the exact area of \(R\). (Solutions based entirely on graphical or numerical methods are not acceptable.)

The diagram below shows a curve with equation \(y = (x + 2)(x - 2)(x + 1)\). \includegraphics{figure_16} Calculate the total area of the two shaded regions. [8]

The diagram below shows a sketch of the curve \(C_1\) with equation \(y = -x^2 + \pi x + 1\) and a sketch of the curve \(C_2\) with equation \(y = \cos 2x\). The curves intersect at the points where \(x = 0\) and \(x = \pi\). \includegraphics{figure_9} Calculate the area of the shaded region enclosed by \(C_1\), \(C_2\) and the \(x\)-axis. Give your answer in terms of \(\pi\). [9]