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# X2 T05 01 discs & washers (2011)

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### X2 T05 01 discs & washers (2011)

1. 1. Volumes of SolidsVolumes By Discs & Washers slice  rotation 
2. 2. Volumes of SolidsVolumes By Discs & Washers slice  rotation About the x axis
3. 3. Volumes of SolidsVolumes By Discs & Washers slice  rotation About the x axis y  f x y x
4. 4. Volumes of SolidsVolumes By Discs & Washers slice  rotation About the x axis y  f x y a b x
5. 5. Volumes of SolidsVolumes By Discs & Washers slice  rotation About the x axis y  f x y a x b x
6. 6. Volumes of SolidsVolumes By Discs & Washers slice  rotation About the x axis y  f x y y a x b x x
7. 7. Volumes of SolidsVolumes By Discs & Washers slice  rotation About the x axis y  f x y y  f x a x b x x
8. 8. Volumes of SolidsVolumes By Discs & Washers slice  rotation About the x axis y  f x y y  f x A x     f  x  2 a x b x x
9. 9. Volumes of SolidsVolumes By Discs & Washers slice  rotation About the x axis y  f x y y  f x A x     f  x  2 a x b x V    f  x   x 2 x
10. 10. Volumes of SolidsVolumes By Discs & Washers slice  rotation About the x axis y  f x y y  f x A x     f  x  2 a x b x V    f  x   x 2 x b V  lim    f  x   x 2 x 0 xa
11. 11. Volumes of SolidsVolumes By Discs & Washers slice  rotation About the x axis y  f x y y  f x A x     f  x  2 a x b x V    f  x   x 2 x b V  lim    f  x   x 2 x 0 xa b     f  x  dx 2 a
12. 12. About the y axis y  f x y x
13. 13. About the y axis y  f x y d c x
14. 14. About the y axis y  f x y d y c x
15. 15. About the y axis y  f x y d y x y c x
16. 16. About the y axis y  f x y d y x  g y y c x
17. 17. About the y axis y  f x y d y x  g y y c A y    g  y  2 x
18. 18. About the y axis y  f x y d y x  g y y c A y    g  y  2 x V   g  y   y 2
19. 19. About the y axis y  f x y d y x  g y y c A y    g  y  2 x V   g  y   y 2 d V  lim   g  y   y 2 y  0 y c
20. 20. About the y axis y  f x y d y x  g y y c A y    g  y  2 x V   g  y   y 2 d V  lim   g  y   y 2 y  0 y c d    g  y  dy 2 c
21. 21. e.g. i  Find the volume generated when the area between y  4  x 2 and the x axis is rotated about the x axis
22. 22. e.g. i  Find the volume generated when the area between y  4  x 2 and the x axis is rotated about the x axis y 4 y  4  x2 -2 2 x
23. 23. e.g. i  Find the volume generated when the area between y  4  x 2 and the x axis is rotated about the x axis y 4 y  4  x2 -2 x 2 x
24. 24. e.g. i  Find the volume generated when the area between y  4  x 2 and the x axis is rotated about the x axis y 4 y  4  x2 -2 x 2 xy  4  x2 x
25. 25. e.g. i  Find the volume generated when the area between y  4  x 2 and the x axis is rotated about the x axis y 4 y  4  x2 -2 x 2 xy  4  x2 A x    4  x  2 2 x
26. 26. e.g. i  Find the volume generated when the area between y  4  x 2 and the x axis is rotated about the x axis y 4 y  4  x2 -2 x 2 xy  4  x2 A x    4  x  2 2 V   16  8 x 2  x 4  x x
27. 27. e.g. i  Find the volume generated when the area between y  4  x 2 and the x axis is rotated about the x axis V  lim   16  8 x 2  x 4  x 2 y 4 x 0 x  2 y  4 x 2 -2 x 2 xy  4  x2 A x    4  x  2 2 V   16  8 x 2  x 4  x x
28. 28. e.g. i  Find the volume generated when the area between y  4  x 2 and the x axis is rotated about the x axis V  lim   16  8 x 2  x 4  x 2 y 4 x 0 x  2 y  4 x 2 2  2  16  8 x 2  x 4 dx -2 x 2 x 0y  4  x2 A x    4  x  2 2 V   16  8 x 2  x 4  x x
29. 29. e.g. i  Find the volume generated when the area between y  4  x 2 and the x axis is rotated about the x axis V  lim   16  8 x 2  x 4  x 2 y 4 x 0 x  2 y  4 x 2 2  2  16  8 x 2  x 4 dx -2 x 2 x 0 2 16 x  8 x 3  1 x 5   2   3 5 0 y  4  x2 A x    4  x  2 2 V   16  8 x 2  x 4  x x
30. 30. e.g. i  Find the volume generated when the area between y  4  x 2 and the x axis is rotated about the x axis V  lim   16  8 x 2  x 4  x 2 y 4 x 0 x  2 y  4 x 2 2  2  16  8 x 2  x 4 dx -2 x 2 x 0 2 16 x  8 x 3  1 x 5   2   3 5 0  32 64 32 0  2       3 5 y  4  x2 512  units 3 A x    4  x  2 2 15 V   16  8 x 2  x 4  x x
31. 31. ii  Find the volume generated when the area enclosed by y  4  x 2 and y  3 is rotated about the line y  3
32. 32. ii  Find the volume generated when the area enclosed by y  4  x 2 and y  3 is rotated about the line y  3 y 4 y3 x y  4  x2
33. 33. ii  Find the volume generated when the area enclosed by y  4  x 2 and y  3 is rotated about the line y  3 y 4 (-1,3) (1,3) y3 x y  4  x2
34. 34. ii  Find the volume generated when the area enclosed by y  4  x 2 and y  3 is rotated about the line y  3 y 4 (-1,3) (1,3) y3 x x y  4  x2
35. 35. ii  Find the volume generated when the area enclosed by y  4  x 2 and y  3 is rotated about the line y  3 y 4 (-1,3) (1,3) y3 x x y  4  x2 x
36. 36. ii  Find the volume generated when the area enclosed by y  4  x 2 and y  3 is rotated about the line y  3 y 4 (-1,3) (1,3) y3 x x y  4  x2 y  3  4  x2  3  1 x2 x
37. 37. ii  Find the volume generated when the area enclosed by y  4  x 2 and y  3 is rotated about the line y  3 y 4 (-1,3) (1,3) y3 x x y  4  x2 y  3  4  x2  3  1 x2 A x    1  x  2 2 x
38. 38. ii  Find the volume generated when the area enclosed by y  4  x 2 and y  3 is rotated about the line y  3 y 4 (-1,3) (1,3) y3 x x y  4  x2 y  3  4  x2  3  1 x2 A x    1  x  2 2V   1  2 x 2  x 4  x x
39. 39. ii  Find the volume generated when the area enclosed by y  4  x 2 and y  3 is rotated about the line y  3  1  2 x 2  x 4  x 1 y 4 V  lim x  0  x  1 (-1,3) (1,3) y3 x x y  4  x2 y  3  4  x2  3  1 x2 A x    1  x  2 2V   1  2 x 2  x 4  x x
40. 40. ii  Find the volume generated when the area enclosed by y  4  x 2 and y  3 is rotated about the line y  3  1  2 x 2  x 4  x 1 y 4 V  lim x  0  x  1 (-1,3) (1,3) y3 1 x  2  1  2 x 2  x 4 dx 0 x y  4  x2 y  3  4  x2  3  1 x2 A x    1  x  2 2V   1  2 x 2  x 4  x x
41. 41. ii  Find the volume generated when the area enclosed by y  4  x 2 and y  3 is rotated about the line y  3  1  2 x 2  x 4  x 1 y 4 V  lim x  0  x  1 (-1,3) (1,3) y3 1 x  2  1  2 x 2  x 4 dx 0 x 1  x  2 x3  1 x5   2  y  4  x2  3 5 0  y  3  4  x2  3  1 x2 A x    1  x  2 2V   1  2 x 2  x 4  x x
42. 42. ii  Find the volume generated when the area enclosed by y  4  x 2 and y  3 is rotated about the line y  3  1  2 x 2  x 4  x 1 y 4 V  lim x  0  x  1 (-1,3) (1,3) y3 1 x  2  1  2 x 2  x 4 dx 0 x 1  x  2 x3  1 x5   2  y  4  x2  3 5 0  y  3  4  x2  3 1 2 1 0  2       3 5   1 x2 16  units 3 A x    1  x  2 2 15V   1  2 x 2  x 4  x x
43. 43. iii  The area between the curves y  x 2 and x  y 2 is rotated about the line y axis. Find the volume generated.
44. 44. iii  The area between the curves y  x 2 and x  y 2 is rotated about the line y axis. Find the volume generated. y y  x2 x  y2 x
45. 45. iii  The area between the curves y  x 2 and x  y 2 is rotated about the line y axis. Find the volume generated. y y  x2 x  y2 (1,1) x
46. 46. iii  The area between the curves y  x 2 and x  y 2 is rotated about the line y axis. Find the volume generated. y y  x2 x  y2 (1,1) y x
47. 47. iii  The area between the curves y  x 2 and x  y 2 is rotated about the line y axis. Find the volume generated. y y  x2 x  y2 (1,1) y x
48. 48. iii  The area between the curves y  x 2 and x  y 2 is rotated about the line y axis. Find the volume generated. y y  x2 x  y2 (1,1) y x 1 x y 2
49. 49. iii  The area between the curves y  x 2 and x  y 2 is rotated about the line y axis. Find the volume generated. y y  x2 x  y2 (1,1) y x 1x y 2 x y 2
50. 50. iii  The area between the curves y  x 2 and x  y 2 is rotated about the line y axis. Find the volume generated. y y  x2 x  y2 (1,1) y x 1x y 2 x y 2  1  2 A y     y    y    2 2 2     
51. 51. iii  The area between the curves y  x 2 and x  y 2 is rotated about the line y axis. Find the volume generated. y y  x2 x  y2 (1,1) y x 1x y 2 x y 2  1  2 A y     y    y    2 2 2      V    y  y 4  y
52. 52. iii  The area between the curves y  x 2 and x  y 2 is rotated about the line y axis. Find the volume generated. y y  x2 V  lim    y  y 4  y 1 x y 2 y 0 y 0 (1,1) y x 1x y 2 x y 2  1  2 A y     y    y    2 2 2      V    y  y 4  y
53. 53. iii  The area between the curves y  x 2 and x  y 2 is rotated about the line y axis. Find the volume generated. y y  x2 V  lim    y  y 4  y 1 x y 2 y 0 y 0 (1,1) 1 y     y  y 4 dy 0 x 1x y 2 x y 2  1  2 A y     y    y    2 2 2      V    y  y 4  y
54. 54. iii  The area between the curves y  x 2 and x  y 2 is rotated about the line y axis. Find the volume generated. y y  x2 V  lim    y  y 4  y 1 x y 2 y 0 y 0 (1,1) 1 y     y  y 4 dy 0 x 1 1 2 1 5   y  y  1 2 5 0x y 2 x y 2  1  2 A y     y    y    2 2 2      V    y  y 4  y
55. 55. iii  The area between the curves y  x 2 and x  y 2 is rotated about the line y axis. Find the volume generated. y y  x2 V  lim    y  y 4  y 1 x y 2 y 0 y 0 (1,1) 1 y     y  y 4 dy 0 x 1 1 2 1 5   y  y  1 2 5 0x y 2 x y 2  1 1 0     2 5  3  1  2   units 3A y     y    y    2 2 2 10      V    y  y 4  y
56. 56. (iv) (1996) y 1 x  y2  0 S 1 4 x -1The shaded region is bounded by the lines x = 1, y = 1 and y = -1 andthe curve x  y  0 . The region is rotated through 360 about the line 2x = 4 to form a solid.When the region is rotated, the line segment S at height y sweeps outan annulus.
57. 57. a) Show that the area of the annulus at height y is equal to   y 4  8 y 2  7 
58. 58. a) Show that the area of the annulus at height y is equal to   y 4  8 y 2  7  y
59. 59. a) Show that the area of the annulus at height y is equal to   y 4  8 y 2  7  y r
60. 60. a) Show that the area of the annulus at height y is equal to   y 4  8 y 2  7  R y
61. 61. 1x  y2  0 S 1 4 x -1
62. 62. 1x  y2  0 S r 1 4 x -1
63. 63. 1x  y2  0 S r  4 1 3 1 4 x -1
64. 64. 1x  y2  0 S r  4 1 3 1 4 x R -1
65. 65. 1x  y2  0 S r  4 1 3 1 4 x R  4 x -1  4  y2
66. 66. a) Show that the area of the annulus at height y is equal to   y 4  8 y 2  7  y
67. 67. a) Show that the area of the annulus at height y is equal to   y 4  8 y 2  7  R  4 x  4  y2 y
68. 68. a) Show that the area of the annulus at height y is equal to   y 4  8 y 2  7  R  4 x  4  y2 y r  4 1 3
69. 69. a) Show that the area of the annulus at height y is equal to   y 4  8 y 2  7  R  4 x  4  y2 y r  4 1  A y    4  y   3  2 2 2 3
70. 70. a) Show that the area of the annulus at height y is equal to   y 4  8 y 2  7  R  4 x  4  y2 y r  4 1  A y    4  y   3  2 2 2 3   16  8 y 2  y 4  9    y 4  8 y 2  7
71. 71. b) Hence find the volume of the solid.
72. 72. b) Hence find the volume of the solid. V    y 4  8 y 2  7  y
73. 73. b) Hence find the volume of the solid. V    y 4  8 y 2  7  y   y 4  8 y 2  7  y 1 V  lim y  0  y  1
74. 74. b) Hence find the volume of the solid. V    y 4  8 y 2  7  y   y 4  8 y 2  7  y 1 V  lim y  0  y  1 1  2   y 4  8 y 2  7 dy 0
75. 75. b) Hence find the volume of the solid. V    y 4  8 y 2  7  y   y 4  8 y 2  7  y 1 V  lim y  0  y  1 1  2   y 4  8 y 2  7 dy 0 1  2  y 5  y 3  7 y  1 8 5  3 0 
76. 76. b) Hence find the volume of the solid. V    y 4  8 y 2  7  y   y 4  8 y 2  7  y 1 V  lim y  0  y  1 1  2   y 4  8 y 2  7 dy 0 1  2  y 5  y 3  7 y  1 8 5  3 0  1  8     2  7 0 5 3  296  units 3 15
77. 77. b) Hence find the volume of the solid. V    y 4  8 y 2  7  y   y 4  8 y 2  7  y 1 V  lim y  0  y  1 1  2   y 4  8 y 2  7 dy Exercise 3A; 0 1 2, 5, 7, 9, 13, 14, 15  2  y 5  y 3  7 y  1 8 5  3 0  Exercise 3B; 1  8     2  1, 2, 5, 7, 9, 10, 11, 12 7 0 5 3  296  units 3 15