11X1 T17 04 areas

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<ul><li> 1. (1) Area Below x axis Areasy y = f(x) x </li></ul> <p> 2. (1) Area Below x axisAreasyy = f(x) A1 ab x 3. (1) Area Below x axisAreasyy = f(x) A1 ab x f x dx 0ba 4. (1) Area Below x axisAreasyy = f(x) A1 ab x f x dx 0b a A1 f x dxba 5. (1) Area Below x axisAreasy y = f(x) A1c x ab A2 f x dx 0b a A1 f x dxba 6. (1) Area Below x axisAreasyy = f(x) A1cx ab A2 f x dx 0c f x dx 0 b ab A1 f x dx b a 7. (1) Area Below x axisAreasyy = f(x) A1cx ab A2 f x dx 0c f x dx 0 b ab A2 f x dxc A1 f x dx b a b 8. Area below x axis is given by; 9. Area below x axis is given by; A f x dx cb 10. Area below x axis is given by; A f x dxc bOR f x dxc b 11. Area below x axis is given by; A f x dx c bOR f x dx c b OR f x dx bc 12. e.g. (i)y y x3-1 1 x 13. e.g. (i)y y x301A x dx x 3 dx 310-1 1 x 14. e.g. (i)y y x30 1A x dx x 3 dx 31 0 4x 1 4 x 01 40 1 41 -1 1 x 15. e.g. (i)y y x301A x dx x 3 dx 310 4 x 1 4 x 0 1 40 1 41 -1 1 x 1 4 4 0 1 4 0 1 411 1 4 4 1 units 2 2 16. e.g. (i)y y x301A x dx x 3 dx 310 4 x 1 4 x 0 1 40 1 41 -1 1 x 1 4 4 0 1 4 0 1 411 1 OR using symmetry of odd function 4 4 1 units 2 2 17. e.g. (i) yy x301A x dx x 3 dx 310 4 x 1 4 x 0 1 40 1 41 -1 1 x 1 4 4 0 1 4 0 1 411 1 OR using symmetry of odd function 4 4 11 A 2 x 3 dx units 202 18. e.g. (i) yy x301A x dx x 3 dx 310 4 x 1 4 x 0 1 40 1 41 -1 1 x 1 4 4 0 1 4 0 1 411 1 OR using symmetry of odd function 4 4 11 A 2 x 3 dx units 202 x 01 412 19. e.g. (i) yy x301A x dx x 3 dx 310 4 x 1 4 x 0 1 40 1 41 -1 1 x 1 4 4 0 1 4 0 1 411 1 OR using symmetry of odd function 4 4 11 A 2 x 3 dx units 202 x 01 412 4 01121 units 22 20. (ii) y y x x 1 x 2 -2 -1 x 21. (ii) yy x x 1 x 2 -2-1xA x 3 x 2 x dx x 3 3 x 2 2 x dx10 3221 22. (ii) yy x x 1 x 2 -2-1xA x 3 x 2 x dx x 3 3 x 2 2 x dx10 322 111 x 4 x3 x 2 x 4 x3 x 2 1 14 2 4 0 23. (ii) yy x x 1 x 2 -2-1xA x 3 x 2 x dx x 3 3 x 2 2 x dx10 322 111 x 4 x3 x 2 x 4 x3 x 2 114 2 4 0 2 1 14 13 12 1 2 4 2 3 2 2 0 4 4 1 units 22 24. (2) Area On The y axisyy = f(x)(b,d)(a,c) x 25. (2) Area On The y axis (1) Make x the subjectyi.e. x = g(y)y = f(x)(b,d)(a,c) x 26. (2) Area On The y axis (1) Make x the subjectyi.e. x = g(y)y = f(x)(2) Substitute the y coordinates(b,d)(a,c) x 27. (2) Area On The y axis (1) Make x the subjectyi.e. x = g(y)y = f(x)(2) Substitute the y coordinates(b,d)d3 A g y dy c(a,c) x 28. (2) Area On The y axis(1) Make x the subject yi.e. x = g(y)y = f(x) (2) Substitute the y coordinates(b,d) d 3 A g y dyc(a,c)xe.g.yy x41 2 x 29. (2) Area On The y axis(1) Make x the subject yi.e. x = g(y)y = f(x) (2) Substitute the y coordinates(b,d) d 3 A g y dyc(a,c)xe.g.yy x4 1x y 41 2 x 30. (2) Area On The y axis(1) Make x the subject yi.e. x = g(y)y = f(x) (2) Substitute the y coordinates(b,d)d 3 A g y dy c(a,c)16 1x A y dy 4 1e.g.yy x4 1x y 41 2 x 31. (2) Area On The y axis(1) Make x the subject yi.e. x = g(y)y = f(x) (2) Substitute the y coordinates(b,d)d 3 A g y dy c(a,c)16 1 xA y dy 4 1 5 16 e.g.yyx4 4 1 y 4 5 1x y 41 2 x 32. (2) Area On The y axis(1) Make x the subject yi.e. x = g(y)y = f(x) (2) Substitute the y coordinates(b,d)d 3 A g y dy c(a,c)16 1 xA y dy 4 1 5 16 e.g.yyx4 4 1 y 4 5 1x y 4 4 5 5 16 4 14 51 2 x 124 units 2 5 33. (3) Area Between Two Curvesy x 34. (3) Area Between Two Curvesyy = f(x)(1) x 35. (3) Area Between Two Curvesyy = g(x)(2) y = f(x)(1)x 36. (3) Area Between Two Curvesyy = g(x)(2) y = f(x)(1)abx 37. (3) Area Between Two Curvesy y = g(x)(2) y = f(x)(1) a bx Area = Area under (1) Area under (2) 38. (3) Area Between Two Curvesyy = g(x)(2) y = f(x)(1) ab x Area = Area under (1) Area under (2) b b f x dx g x dx a a 39. (3) Area Between Two Curvesy y = g(x)(2) y = f(x)(1) a b x Area = Area under (1) Area under (2) bb f x dx g x dx aa b f x g x dx a 40. e.g. Find the area enclosed between the curves y x 5 and y xin the positive quadrant. 41. e.g. Find the area enclosed between the curves y x 5 and y xin the positive quadrant. y y x5yxx 42. e.g. Find the area enclosed between the curves y x 5 and y xin the positive quadrant. y y x5yxxx x 5 43. e.g. Find the area enclosed between the curves y x 5 and y xin the positive quadrant. y y x5yx xx x 5 x5 x 0xx 4 1 0x 0 or x 1 44. e.g. Find the area enclosed between the curves y x 5 and y xin the positive quadrant. y y x5yx A x x 5 dx 1x 0x x 5 x5 x 0xx 4 1 0x 0 or x 1 45. e.g. Find the area enclosed between the curves y x 5 and y xin the positive quadrant. y y x5yx A x x 5 dx 1x 0x x 5 111 x5 x 0 x2 x6 26 0xx 4 1 0x 0 or x 1 46. e.g. Find the area enclosed between the curves y x 5 and y xin the positive quadrant. y y x5yx A x x 5 dx 1x 0x x 5 111 x5 x 0 x2 x6 26 0xx 4 1 0 1 1 2 1 1 6 0 x 0 or x 1 2 6 1 unit 2 3 47. 2002 HSC Question 4d) The graphs of y x 4 and y x 2 4 x intersect at the points 4,0 A. 48. 2002 HSC Question 4d)The graphs of y x 4 and y x 2 4 x intersect at the points 4,0 A. (i) Find the coordinates of A (2) 49. 2002 HSC Question 4d)The graphs of y x 4 and y x 2 4 x intersect at the points 4,0 A. (i) Find the coordinates of A (2) To find points of intersection, solve simultaneously 50. 2002 HSC Question 4d)The graphs of y x 4 and y x 2 4 x intersect at the points 4,0 A. (i) Find the coordinates of A (2) To find points of intersection, solve simultaneously x 4 x2 4x 51. 2002 HSC Question 4d)The graphs of y x 4 and y x 2 4 x intersect at the points 4,0 A. (i) Find the coordinates of A (2) To find points of intersection, solve simultaneously x 4 x2 4xx2 5x 4 0 52. 2002 HSC Question 4d)The graphs of y x 4 and y x 2 4 x intersect at the points 4,0 A. (i) Find the coordinates of A (2) To find points of intersection, solve simultaneously x 4 x2 4x x2 5x 4 0 x 4 x 1 0 53. 2002 HSC Question 4d)The graphs of y x 4 and y x 2 4 x intersect at the points 4,0 A. (i) Find the coordinates of A (2) To find points of intersection, solve simultaneously x 4 x2 4x x2 5x 4 0 x 4 x 1 0x 1 or x 4 54. 2002 HSC Question 4d)The graphs of y x 4 and y x 2 4 x intersect at the points 4,0 A. (i) Find the coordinates of A (2) To find points of intersection, solve simultaneously x 4 x2 4x x2 5x 4 0 x 4 x 1 0x 1 or x 4 A is (1, 3) 55. (ii) Find the area of the shaded region bounded by y x 2 4 x and (3)y x 4. 56. (ii) Find the area of the shaded region bounded by y x 2 4 x and (3)y x 4.4 A x 4 x 2 4 x dx1 57. (ii) Find the area of the shaded region bounded by y x 2 4 x and (3)y x 4.4 A x 4 x 2 4 x dx14 x 2 5 x 4 dx1 58. (ii) Find the area of the shaded region bounded by y x 2 4 x and (3)y x 4.4 A x 4 x 2 4 x dx14 x 2 5 x 4 dx1 4 1 x3 5 x 2 4 x 32 1 59. (ii) Find the area of the shaded region bounded by y x 2 4 x and(3)y x 4.4 A x 4 x 2 4 x dx14 x 2 5 x 4 dx1 4 1 x3 5 x 2 4 x 32 1 1 3 5 2 1 3 5 2 4 4 4 4 1 1 4 1 3 23 2 60. (ii) Find the area of the shaded region bounded by y x 2 4 x and(3)y x 4. 4A x 4 x 2 4 x dx 1 4 x 2 5 x 4 dx 1 4 1 x3 5 x 2 4 x 32 1 1 3 5 2 1 3 5 2 4 4 4 4 1 1 4 1 3 23 2 9 units 2 2 61. 2005 HSC Question 8b)(3) The shaded region in the diagram is bounded by the circle of radius 2 centred at the origin, the parabola y x 2 3 x 2 , and the x axis. By considering the difference of two areas, find the area of the shaded region. 62. 2005 HSC Question 8b)(3) The shaded region in the diagram is bounded by the circle of radius 2 centred at the origin, the parabola y x 2 3 x 2 , and the x axis. By considering the difference of two areas, find the area of the shaded region. Note: area must be broken up into two areas, due to the differentboundaries. 63. 2005 HSC Question 8b)(3) The shaded region in the diagram is bounded by the circle of radius 2 centred at the origin, the parabola y x 2 3 x 2 , and the x axis. By considering the difference of two areas, find the area of the shaded region. Note: area must be broken up into two areas, due to the differentboundaries. 64. 2005 HSC Question 8b)(3) The shaded region in the diagram is bounded by the circle of radius 2 centred at the origin, the parabola y x 2 3 x 2 , and the x axis. By considering the difference of two areas, find the area of the shaded region. Note: area must be broken up into two areas, due to the differentboundaries. Area between circle and parabola 65. 2005 HSC Question 8b)(3) The shaded region in the diagram is bounded by the circle of radius 2 centred at the origin, the parabola y x 2 3 x 2 , and the x axis. By considering the difference of two areas, find the area of the shaded region. Note: area must be broken up into two areas, due to the differentboundaries. Area between circle and parabola and area between circle and x axis 66. It is easier to subtract the area under the parabola from the quadrant. 67. It is easier to subtract the area under the parabola from the quadrant.1 A 2 x 2 3 x 2 dx12 4 0 68. It is easier to subtract the area under the parabola from the quadrant.1 A 2 x 2 3 x 2 dx12 4 01 x x 2x1 3 3 23 20 69. It is easier to subtract the area under the parabola from the quadrant.1 A 2 x 2 3 x 2 dx12 4 01 x x 2x1 3 3 23 20 1 3 3 2 1 1 2 1 0 3 2 70. It is easier to subtract the area under the parabola from the quadrant.1 A 2 x 2 3 x 2 dx1 2 4 01 x x 2x1 3 3 23 20 1 3 3 2 1 1 2 1 0 3 2 5 units 2 6 71. Exercise 11E; 2bceh, 3bd, 4bd, 5bd, 7begj, 8d, 9a, 11, 18*Exercise 11F; 1bdeh, 4bd, 7d, 10, 11b, 13, 15* </p>