Direct integration for area

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.

7 In this question you must show detailed reasoning. \includegraphics[max width=\textwidth, alt={}, center]{1d1e41f3-a834-4230-b6e1-4b0be9450d30-5_643_716_303_242} Find the exact area of the shaded region shown in the diagram, enclosed by the \(x\)-axis and the curve \(y = - 3 x ^ { 2 } + 7 x - 2\).

6 Use integration to show that the area bounded by the \(x\)-axis and the curve with equation \(y = ( x - 1 ) ^ { 2 } ( x - 3 )\) is \(\frac { 4 } { 3 }\) square units.

11 Four finite regions \(A , B , C\) and \(D\) are enclosed by the curve with equation $$y = x ^ { 3 } - 7 x ^ { 2 } + 11 x + 6$$ and the lines \(y = k , x = 1\) and \(x = 4\), as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{1d017497-11b1-4096-b83a-63314188307e-12_865_1056_520_493} The areas of \(B\) and \(C\) are equal.
Find the value of \(k\).

8. \begin{figure}[h] \end{figure} The curve \(C\), shown in Fig. 1, represents the graph of \(y = \frac { x ^ { 2 } } { 25 } , x \geq 0\).
The points \(A\) and \(B\) on the curve \(C\) have \(x\)-coordinates 5 and 10 respectively.

1. Write down the \(y\)-coordinates of \(A\) and \(B\).
2. Find an equation of the tangent to \(C\) at \(A\). The finite region \(R\) is enclosed by \(C\), the \(y\)-axis and the lines through \(A\) and \(B\) parallel to the \(x\)-axis.
3. For points \(( x , y )\) on \(C\), express \(x\) in terms of \(y\).
4. Use integration to find the area of \(R\).

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 \(C\).
5. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
6. Find the coordinates of \(Q\).
7. Show that \(P Q\) is parallel to the \(x\)-axis.
8. Calculate the area, shown shaded in Fig. 3, bounded by \(C\) and the line \(P Q\).

7. \begin{figure}[h] \end{figure} Fig. 1 shows part of the curve \(C\) with equation \(y = \frac { 3 } { 2 } x ^ { 2 } - \frac { 1 } { 4 } x ^ { 3 }\).
The curve \(C\) touches the \(x\)-axis at the origin and passes through the point \(A ( p , 0 )\).

1. Show that \(p = 6\).
2. Find an equation of the tangent to \(C\) at \(A\). The curve \(C\) has a maximum at the point \(P\).
3. Find the \(x\)-coordinate of \(P\). The shaded region \(R\), in Fig. 1, is bounded by \(C\) and the \(x\)-axis.
4. Find the area of \(R\).

7. \begin{figure}[h] \end{figure} Figure 3 shows part of the curve \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \frac { 1 - 8 x ^ { 3 } } { x ^ { 2 } } , \quad x \neq 0 .$$

1. Solve the equation \(\mathrm { f } ( x ) = 0\).
2. Find \(\int \mathrm { f } ( x ) \mathrm { d } x\).
3. Find the area of the shaded region bounded by the curve \(y = \mathrm { f } ( x )\), the \(x\)-axis and the line \(x = 2\).

3. Find the area of the finite region enclosed by the curve \(y = 5 x - x ^ { 2 }\) and the \(x\)-axis.

5. \begin{figure}[h] \end{figure} Figure 2 shows the curve with equation \(y = 4 x ^ { \frac { 1 } { 3 } } - x , x \geq 0\).
The curve meets the \(x\)-axis at the origin and at the point \(A\) with coordinates \(( a , 0 )\).

1. Show that \(a = 8\).
2. Find the area of the finite region bounded by the curve and the positive \(x\)-axis.

4. $$f ( x ) = 2 - x - x ^ { 3 }$$

1. Show that \(\mathrm { f } ( x )\) is decreasing for all values of \(x\).
2. Verify that the point \(( 1,0 )\) lies on the curve \(y = \mathrm { f } ( x )\).
3. Find the area of the region bounded by the curve \(y = \mathrm { f } ( x )\) and the coordinate axes.

3. \begin{figure}[h] \end{figure} Figure 1 shows a circle with radius \(r\) and centre at the origin.
The region \(R\), shown shaded in Figure 1, is bounded by the \(x\)-axis and the part of the circle for which \(y > 0\) The region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis to create a sphere with volume \(V\) Use integration to show that \(V = \frac { 4 } { 3 } \pi r ^ { 3 }\)

8 In this question you must show detailed reasoning. The diagram shows part of the graph of \(y = 2 x ^ { \frac { 1 } { 3 } } - \frac { 7 } { x ^ { \frac { 1 } { 3 } } }\). The shaded region is enclosed by the curve, the \(x\)-axis and the lines \(x = 8\) and \(x = a\), where \(a > 8\). \includegraphics[max width=\textwidth, alt={}, center]{efde7b10-b4f3-469f-ba91-b765a16ea835-5_577_1164_477_438} Given that the area of the shaded region is 45 square units, find the value of \(a\).

3 In this question you must show detailed reasoning. The diagram shows the curve with equation \(y = x ^ { 5 }\) and the square \(O A B C\) where the points \(A , B\) and \(C\) have coordinates \(( 1,0 ) , ( 1,1 )\) and \(( 0,1 )\) respectively. The curve cuts the square into two parts. \includegraphics[max width=\textwidth, alt={}, center]{60e1e785-c34b-48ef-a63f-13a25fee186e-04_658_780_1318_230} Show that the relationship between the areas of the two parts of the square is \(\frac { \text { Area to left of curve } } { \text { Area below curve } } = 5\).

8 In this question you must show detailed reasoning. Fig. 8 shows the graph of a quadratic function. The graph crosses the axes at the points \(( - 1,0 ) , ( 0 , - 4 )\) and \(( 2,0 )\). \begin{figure}[h] \end{figure} Find the area of the finite region bounded by the curve and the \(x\)-axis.

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.

10 \includegraphics[max width=\textwidth, alt={}, center]{fbe396af-4d5b-4c3f-b528-b2e0783c7bc4-13_696_931_269_607} The diagram shows the graph of \(y = 1 - 3 x ^ { - \frac { 1 } { 2 } }\).

1. Verify that the curve intersects the \(x\)-axis at \(( 9,0 )\).
2. The shaded region is enclosed by the curve, the \(x\)-axis and the line \(x = a\) (where \(a > 9\) ). Given that the area of the shaded region is 4 square units, find the value of \(a\). June 2007 1 A geometric progression \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 15 \quad \text { and } \quad u _ { n + 1 } = 0.8 u _ { n } \text { for } n \geqslant 1 \text {. }$$
3. Write down the values of \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\).
4. Find \(\sum _ { n = 1 } ^ { 20 } u _ { n }\). 2 Expand \(\left( x + \frac { 2 } { x } \right) ^ { 4 }\) completely, simplifying the terms. 3 Use logarithms to solve the equation \(3 ^ { 2 x + 1 } = 5 ^ { 200 }\), giving the value of \(x\) correct to 3 significant figures. 4 \includegraphics[max width=\textwidth, alt={}, center]{fbe396af-4d5b-4c3f-b528-b2e0783c7bc4-14_543_855_1155_646} The diagram shows the curve \(\mathrm { y } = \sqrt { 4 \mathrm { x } + 1 }\).
5. Use the trapezium rule, with strips of width 0.5 , to find an approximate value for the area of the region bounded by the curve \(\mathrm { y } = \sqrt { 4 \mathrm { x } + 1 }\), the x -axis, and the lines \(\mathrm { x } = 1\) and \(\mathrm { x } = 3\). Give your answer correct to 3 significant figures.
6. State with a reason whether this approximation is an under-estimate or an over-estimate. 5
7. Show that the equation $$3 \cos ^ { 2 } \theta = \sin \theta + 1$$ can be expressed in the form $$3 \sin ^ { 2 } \theta + \sin \theta - 2 = 0 .$$
8. Hence solve the equation $$3 \cos ^ { 2 } \theta = \sin \theta + 1 ,$$ giving all values of \(\theta\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\). 6 (a) (i) Find \(\int x \left( x ^ { 2 } - 4 \right) d x\)
9. Hence evaluate \(\int _ { 1 } ^ { 6 } x \left( x ^ { 2 } - 4 \right) d x\).
   (b) Find \(\int \frac { 6 } { x ^ { 3 } } d x\) 7 (a) In an arithmetic progression, the first term is 12 and the sum of the first 70 terms is 12915 . Find the common difference.
   (b) In a geometric progression, the second term is - 4 and the sum to infinity is 9 . Find the common ratio. 8 \includegraphics[max width=\textwidth, alt={}, center]{fbe396af-4d5b-4c3f-b528-b2e0783c7bc4-15_305_744_1043_703} The diagram shows a triangle ABC , where angle BAC is 0.9 radians. BAD is a sector of the circle with centre \(A\) and radius \(A B\).
10. The area of the sector \(B A D\) is \(16.2 \mathrm {~cm} ^ { 2 }\). Show that the length of \(A B\) is 6 cm .
11. The area of triangle \(A B C\) is twice the area of sector \(B A D\). Find the length of \(A C\).
12. Find the perimeter of the region \(B C D\). 9 The polynomial \(f ( x )\) is given by $$f ( x ) = x ^ { 3 } + 6 x ^ { 2 } + x - 4$$
13. (a) Show that \(( x + 1 )\) is a factor of \(f ( x )\).
    (b) Hence find the exact roots of the equation \(f ( x ) = 0\).
14. (a) Show that the equation $$2 \log _ { 2 } ( x + 3 ) + \log _ { 2 } x - \log _ { 2 } ( 4 x + 2 ) = 1$$ can be written in the form \(\mathrm { f } ( \mathrm { x } ) = 0\).
    (b) Explain why the equation $$2 \log _ { 2 } ( x + 3 ) + \log _ { 2 } x - \log _ { 2 } ( 4 x + 2 ) = 1$$ has only one real root and state the exact value of this root. \section*{Jan 2008} 1 The diagram shows a sector \(A O B\) of a circle with centre \(O\) and radius 11 cm . The angle \(A O B\) is 0.7 radians. Find the area of the segment shaded in the diagram. 2 Use the trapezium rule, with 3 strips each of width 2, to estimate the value of $$\int _ { 1 } ^ { 7 } \sqrt { x ^ { 2 } + 3 } \mathrm {~d} x$$ 3 Express each of the following as a single logarithm:
15. \(\log _ { a } 2 + \log _ { a } 3\),
16. \(2 \log _ { 10 } x - 3 \log _ { 10 } y\). 4 \includegraphics[max width=\textwidth, alt={}, center]{fbe396af-4d5b-4c3f-b528-b2e0783c7bc4-16_515_713_1567_715} In the diagram, angle \(B D C = 50 ^ { \circ }\) and angle \(B C D = 62 ^ { \circ }\). It is given that \(A B = 10 \mathrm {~cm} , A D = 20 \mathrm {~cm}\) and \(B C = 16 \mathrm {~cm}\).
17. Find the length of \(B D\).
18. Find angle \(B A D\). 5 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 12 \sqrt { x }\). The curve passes through the point (4,50). Find the equation of the curve. 6 A sequence of terms \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { n } = 2 n + 5 , \quad \text { for } n \geqslant 1$$
19. Write down the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
20. State what type of sequence it is.
21. Given that \(\sum _ { n = 1 } ^ { N } u _ { n } = 2200\), find the value of \(N\). 7 \includegraphics[max width=\textwidth, alt={}, center]{fbe396af-4d5b-4c3f-b528-b2e0783c7bc4-17_588_569_854_788} The diagram shows part of the curve \(y = x ^ { 2 } - 3 x\) and the line \(x = 5\).
22. 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.
23. Use integration to find the exact total area of the shaded regions. 8 The first term of a geometric progression is 10 and the common ratio is 0.8.
24. Find the fourth term.
25. Find the sum of the first 20 terms, giving your answer correct to 3 significant figures.
26. The sum of the first \(N\) terms is denoted by \(S _ { N }\), and the sum to infinity is denoted by \(S _ { \infty }\). Show that the inequality \(S _ { \infty } - S _ { N } < 0.01\) can be written as $$0.8 ^ { N } < 0.0002$$ and use logarithms to find the smallest possible value of \(N\). 9
27. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{fbe396af-4d5b-4c3f-b528-b2e0783c7bc4-18_378_770_274_731} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} Fig. 1 shows the curve \(y = 2 \sin x\) for values of \(x\) such that \(- 180 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\). State the coordinates of the maximum and minimum points on this part of the curve.
28. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{fbe396af-4d5b-4c3f-b528-b2e0783c7bc4-18_378_771_954_730} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} Fig. 2 shows the curve \(y = 2 \sin x\) and the line \(y = k\). The smallest positive solution of the equation \(2 \sin x = k\) is denoted by \(\alpha\). State, in terms of \(\alpha\), and in the range \(- 180 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\),
    (a) another solution of the equation \(2 \sin x = k\),
    (b) one solution of the equation \(2 \sin x = - k\).
29. Find the \(x\)-coordinates of the points where the curve \(y = 2 \sin x\) intersects the curve \(y = 2 - 3 \cos ^ { 2 } x\), for values of \(x\) such that \(- 180 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\). 10
30. Find the binomial expansion of \(( 2 x + 5 ) ^ { 4 }\), simplifying the terms.
31. Hence show that \(( 2 x + 5 ) ^ { 4 } - ( 2 x - 5 ) ^ { 4 }\) can be written as $$320 x ^ { 3 } + k x$$ where the value of the constant \(k\) is to be stated.
32. Verify that \(x = 2\) is a root of the equation $$( 2 x + 5 ) ^ { 4 } - ( 2 x - 5 ) ^ { 4 } = 3680 x - 800$$ and find the other possible values of \(x\).

8

1. Show that $$\int _ { 1 } ^ { a } \left( 6 - \frac { 12 } { \sqrt { x } } \right) \mathrm { d } x = 6 a - 24 \sqrt { a } + 18$$ 8
2. The curve \(y = 6 - \frac { 12 } { \sqrt { x } }\), the line \(x = 1\) and the line \(x = a\) are shown in the diagram below. The shaded region \(R _ { 1 }\) is bounded by the curve, the line \(x = 1\) and the \(x\)-axis.
   The shaded region \(R _ { 2 }\) is bounded by the curve, the line \(x = a\) and the \(x\)-axis. \includegraphics[max width=\textwidth, alt={}, center]{9cd7f38d-a2a1-4fd3-9ed9-cb389e8ee3b6-09_705_931_632_648} It is given that the areas of \(R _ { 1 }\) and \(R _ { 2 }\) are equal.
   Find the value of \(a\) Fully justify your answer.

6 A curve has equation \(y = 6 x ^ { 2 } + \frac { 8 } { x ^ { 2 } }\) and is sketched below for \(x > 0\) \includegraphics[max width=\textwidth, alt={}, center]{f2bf5e19-98ba-4047-9023-3cfe20987e01-06_638_842_539_758} Find the area of the region bounded by the curve, the \(x\)-axis and the lines \(x = a\) and \(x = 2 a\), where \(a > 0\), giving your answer in terms of \(a\) [0pt] [4 marks]

2
The graph of \(y = \mathrm { f } ( x )\) intersects the \(x\)-axis at ( \(- 3,0\) ), ( 0,0 ) and ( 2,0 ) as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{28ee1134-b938-4c8f-b4ef-f3f9b210fdef-03_634_885_415_644} The shaded region \(A\) has an area of 189 The shaded region \(B\) has an area of 64
Find the value of \(\int _ { - 3 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x\) Circle your answer.
-253
-125
125
253