11 x1 t16 05 volumes (2013)

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11 x1 t16 05 volumes (2013)

  1. 1. Volumes of Solids of Revolution y y = f(x) x
  2. 2. Volumes of Solids of Revolution y y = f(x) a b x
  3. 3. Volumes of Solids of Revolution y y = f(x) a b x
  4. 4. Volumes of Solids of Revolution y y = f(x) a Volume of a solid of revolution about; b x
  5. 5. Volumes of Solids of Revolution y y = f(x) a b x Volume of a solid of revolution about; i x axis : V   y 2 dx  b a
  6. 6. Volumes of Solids of Revolution y y = f(x) a b x Volume of a solid of revolution about; i x axis : V   y 2 dx  b a ii  y axis : V   c x 2 dy d
  7. 7. e.g. (i) cone y y  mx h x
  8. 8. e.g. (i) cone y y  mx h x
  9. 9. V    y 2 dx e.g. (i) cone h y y  mx h x    m 2 x 2 dx 0
  10. 10. V    y 2 dx e.g. (i) cone h y y  mx h x    m 2 x 2 dx 0 h 2 1 3   m  x  3  0
  11. 11. V    y 2 dx e.g. (i) cone h y y  mx h x    m 2 x 2 dx 0 h 2 1 3   m  x  3  0 1 2 3  m h  0  3 1 2 3  m h 3
  12. 12. V    y 2 dx e.g. (i) cone h y y  mx r h x    m 2 x 2 dx 0 h 2 1 3   m  x  3  0 1 2 3  m h  0  3 1 2 3  m h 3
  13. 13. V    y 2 dx e.g. (i) cone h y y  mx r h x r m h    m 2 x 2 dx 0 h 2 1 3   m  x  3  0 1 2 3  m h  0  3 1 2 3  m h 3
  14. 14. V    y 2 dx e.g. (i) cone h y y  mx r h x r m h    m 2 x 2 dx 0 h 2 1 3   m  x  3  0 1 2 3  m h  0  3 1 2 3  m h 3 2 1 r     h3 3 h r 2 h 3  3h 2 r 2 h  3
  15. 15. (ii) sphere y x2  y2  r 2 rx
  16. 16. (ii) sphere y x2  y2  r 2 rx
  17. 17. (ii) sphere y x2  y2  r 2 V    y 2 dx r  2  r 2  x 2 dx 0 rx
  18. 18. (ii) sphere y x2  y2  r 2 V    y 2 dx r  2  r 2  x 2 dx 0 r rx r 2 x  1 x 3   2  3 0  
  19. 19. (ii) sphere y x2  y2  r 2 V    y 2 dx r  2  r 2  x 2 dx 0 r rx r 2 x  1 x 3   2  3 0   1    2  r 2 r   r 3   0   3    2 3  2  r  3  4  r 3 3
  20. 20. (iii) Find the volume of the solid generated when y  x 2 is revolved around the y axis between y  0 and y  1.
  21. 21. (iii) Find the volume of the solid generated when y  x 2 is revolved around the y axis between y  0 and y  1. y y  x2 x
  22. 22. (iii) Find the volume of the solid generated when y  x 2 is revolved around the y axis between y  0 and y  1. y  x2 y 1 x
  23. 23. (iii) Find the volume of the solid generated when y  x 2 is revolved around the y axis between y  0 and y  1. y  x2 y 1 x
  24. 24. (iii) Find the volume of the solid generated when y  x 2 is revolved around the y axis between y  0 and y  1. y  x2 y V   1  1  ydy 0 x x 2 dy
  25. 25. (iii) Find the volume of the solid generated when y  x 2 is revolved around the y axis between y  0 and y  1. y  x2 y V   1  x 2 dy 1  ydy 0  x  y  2 2 1 0
  26. 26. (iii) Find the volume of the solid generated when y  x 2 is revolved around the y axis between y  0 and y  1. y  x2 y V   1  x 2 dy 1  ydy 0  x    y  2  2  2 2 1 0 1 2  0 units 3
  27. 27. (iv) Find the volume of the solid when the shaded region is rotated about the x axis y y  5x 5 x
  28. 28. (iv) Find the volume of the solid when the shaded region is rotated about the x axis y V    y 2 dx y  5x 1 5 2    52  5 x  dx   0 1 x    25  25 x 2 dx 0
  29. 29. (iv) Find the volume of the solid when the shaded region is rotated about the x axis y V    y 2 dx y  5x 1 5 2    52  5 x  dx   0 1 x    25  25 x 2 dx 0 1  x  1 x3   25   3 0 
  30. 30. (iv) Find the volume of the solid when the shaded region is rotated about the x axis y V    y 2 dx y  5x 1 5 2    52  5 x  dx   0 1 x    25  25 x 2 dx 0 1  x  1 x3   25   3 0  1 1 13 0  25      3  50  unit 3 3
  31. 31. (iv) Find the volume of the solid when the shaded region is rotated about the x axis y V    y 2 dx y  5x 1 5 2    52  5 x  dx   0 1 x OR    25  25 x 2 dx 0 1  x  1 x3   25   3 0  1 1 13 0  25      3  50  unit 3 3
  32. 32. (iv) Find the volume of the solid when the shaded region is rotated about the x axis y V    y 2 dx y  5x 1 5 2    52  5 x  dx   0 1 x 1 OR V  r 2 h  r 2 h 3 2 2   5 1 3 50  units 3 3    25  25 x 2 dx 0 1  x  1 x3   25   3 0  1 1 13 0  25      3  50  unit 3 3
  33. 33. 2005 HSC Question 6c) The graphs of the curves y  x 2 and y  12  2 x 2 are shown in the diagram. (i) Find the points of intersection of the two curves. (1)
  34. 34. 2005 HSC Question 6c) The graphs of the curves y  x 2 and y  12  2 x 2 are shown in the diagram. (i) Find the points of intersection of the two curves. x 2  12  2 x 2 (1)
  35. 35. 2005 HSC Question 6c) The graphs of the curves y  x 2 and y  12  2 x 2 are shown in the diagram. (i) Find the points of intersection of the two curves. x 2  12  2 x 2 3 x 2  12 x2  4 x  2 (1)
  36. 36. 2005 HSC Question 6c) The graphs of the curves y  x 2 and y  12  2 x 2 are shown in the diagram. (i) Find the points of intersection of the two curves. x 2  12  2 x 2 3 x 2  12 x2  4 x  2  meet at  2, 4  (1)
  37. 37. (ii) The shaded region between the two curves and the y axis is (3) rotated about the y axis. By splitting the shaded region into two parts, or otherwise, find the volume of the solid formed.
  38. 38. (ii) The shaded region between the two curves and the y axis is (3) rotated about the y axis. By splitting the shaded region into two parts, or otherwise, find the volume of the solid formed.
  39. 39. (ii) The shaded region between the two curves and the y axis is (3) rotated about the y axis. By splitting the shaded region into two parts, or otherwise, find the volume of the solid formed. V    x 2 dy
  40. 40. (ii) The shaded region between the two curves and the y axis is (3) rotated about the y axis. By splitting the shaded region into two parts, or otherwise, find the volume of the solid formed. V    x 2 dy
  41. 41. (ii) The shaded region between the two curves and the y axis is (3) rotated about the y axis. By splitting the shaded region into two parts, or otherwise, find the volume of the solid formed. x2  y V    x 2 dy
  42. 42. (ii) The shaded region between the two curves and the y axis is (3) rotated about the y axis. By splitting the shaded region into two parts, or otherwise, find the volume of the solid formed. x2  y 1 x  6 y 2 2 V    x 2 dy
  43. 43. (ii) The shaded region between the two curves and the y axis is (3) rotated about the y axis. By splitting the shaded region into two parts, or otherwise, find the volume of the solid formed. x2  y 1 x  6 y 2 2 V    x 2 dy 4 12   1   V   6 y  dy  ydy 2  0 4
  44. 44. (ii) The shaded region between the two curves and the y axis is (3) rotated about the y axis. By splitting the shaded region into two parts, or otherwise, find the volume of the solid formed. x2  y 1 x  6 y 2 2 V    x 2 dy 4 12   1   V   6 y  dy  ydy 2  0 4 4 12 1 2 1 2   6 y  y     y    2 4 4 0  
  45. 45. (ii) The shaded region between the two curves and the y axis is (3) rotated about the y axis. By splitting the shaded region into two parts, or otherwise, find the volume of the solid formed. x2  y 1 x  6 y 2 2 V    x 2 dy 4 12     1   V   6 y  dy  ydy 2  0 4 4 12 1 2 1 2   6 y  y     y    2 4 4 0   1 2  2 2   6  4    4   0  12    4  4 2  
  46. 46. (ii) The shaded region between the two curves and the y axis is (3) rotated about the y axis. By splitting the shaded region into two parts, or otherwise, find the volume of the solid formed. x2  y 1 x  6 y 2 2 V    x 2 dy 4 12     1   V   6 y  dy  ydy 2  0 4 4 12 1 2 1 2   6 y  y     y    2 4 4 0   1 2  2 2   6  4    4   0  12    4  4 2  84 units3  
  47. 47. Exercise 11G; 3bdef, 4eg, 5cd, 6d, 8ad, 12, 16a, 19 to 22, 24*, 25*

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