X2 T06 01 discs and washers (2010)

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X2 T06 01 discs and washers (2010)

  1. 1. Volumes of Solids Volumes By Discs & Washers slice  rotation 
  2. 2. Volumes of Solids Volumes By Discs & Washers slice  rotation  About the x axis
  3. 3. Volumes of Solids Volumes By Discs & Washers slice  rotation  About the x axis y  f x y x
  4. 4. Volumes of Solids Volumes By Discs & Washers slice  rotation  About the x axis y  f x y a b x
  5. 5. Volumes of Solids Volumes By Discs & Washers slice  rotation  About the x axis y  f x y a x b x
  6. 6. Volumes of Solids Volumes By Discs & Washers slice  rotation  About the x axis y  f x y y a x b x x
  7. 7. Volumes of Solids Volumes 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 Solids Volumes 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 Solids Volumes 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 Solids Volumes 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 Solids Volumes 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 x y  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 x y  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 x y  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 x y  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 0 y  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 1 x 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 1 x 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 1 x 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 1 x 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 1 x 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 0 x 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 0 x y 2 x y 2  1 1 0     2 5  3  1  2   units 3 A 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 -1 The shaded region is bounded by the lines x = 1, y = 1 and y = -1 and the curve x  y  0 . The region is rotated through 360 about the line 2 x = 4 to form a solid. When the region is rotated, the line segment S at height y sweeps out an 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. 1 x  y2  0 S 1 4 x -1
  62. 62. 1 x  y2  0 S r 1 4 x -1
  63. 63. 1 x  y2  0 S r  4 1 3 1 4 x -1
  64. 64. 1 x  y2  0 S r  4 1 3 1 4 x R -1
  65. 65. 1 x  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

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