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Surface of Revolution: Formed by Rotating Area Between Two Functions

Surface of Revolution: Formed by Rotating Area Between Two Functions
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2018-01-29 10:13
FunctionCalculus
In collections GeoGebra 3D & AR: PreCalc & Calculus Resources
This example demonstrates how to form a surface of revolution by rotating the area between y=f(x) and y=g(x). Key mathematical concepts include functions, integration, and rotation. GeoGebra commands used include Surface, Textfield, IntegralBetween, Rotate, etc. Suitable for high school and above students learning calculus and geometric transformations.

GeoGebra Commands & Parameters

  1. Surface(u, c + ((f(u) - c) * cos(v)), ((f(u) - c) * sin(v)), u, a, b, v, 0, n) → OuterH
  2. Textfield(c) → InputBox1
  3. Surface(u, c + ((h(u) - c) * cos(v)), ((h(u) - c) * sin(v)), u, a, b, v, 0, n) → InnerH
  4. IntegralBetween(f, h, a, b) → r
  5. Surface(o + ((u - o) * cos(v)), f(u), ((u - o) * sin(v)), u, a, b, v, 0, n) → VerticalOuter
  6. Surface(o + ((u - o) * cos(v)), h(u), ((u - o) * sin(v)), u, a, b, v, 0, n) → VerticalInner
  7. Segment((a, 0), (a, h(a))) → g_1
  8. Segment((b, h(b), 0), (b, f(b), 0)) → h_1
  9. Rotate(h_1, (-n), (b, c, 0), k) → i_1
  10. Rotate(g, (-n), k) → g_2
  11. Rotate(p, (-n), k) → p_1
  12. Segment((a, h(a), 0), (a, f(a), 0)) → j_1
  13. Rotate(j_1, (-n), k) → k_1
  14. CurveCartesian(a, (f(a) * t) + (h(a) * (1 - t)), 0, t, 0, 1) → w
  15. CurveCartesian(b, (f(b) * t) + (h(b) * (1 - t)), 0, t, 0, 1) → b_1
  16. Surface(w, n, k) → c_1
  17. Mirror(c_1, z = 0) → LeftH
  18. Surface(b_1, n, k) → e_1
  19. Mirror(e_1, z = 0) → RightH
  20. Surface(w, n, f_1) → d
  21. Surface(b_1, n, f_1) → e
  22. Mirror(d, z = 0) → LesserBoundaryV
  23. Mirror(e, z = 0) → GreaterBoundaryV
  24. Rotate(p, (-n), f_1) → p_2
  25. Rotate(b_1, (-n), f_1) → u
  26. Rotate(g, (-n), f_1) → g_3
  27. Rotate(w, (-n), f_1) → w'
  28. Rotate(D, If[0 < n ≤ pi, (-n), -3.14], k) → D'
  29. Rotate(E, If[0 < n ≤ pi, (-n), -3.14], k) → E'
  30. CircleArc((b, c, 0), E, E') → q
  31. CircleArc((b, c, 0), D, D') → t
  32. Mirror(E, k) → E'_1
  33. CircleArc(F, E, E'_1) → d_1
  34. CircleArc(F, E, E'_1) → q_1
  35. Mirror(D, k) → D'_1
  36. Rotate(D'_1, If[pi < n ≤ (2 * pi), (-(n - pi)), 6.28], k) → G
  37. CircleArc(F, D', G) → r_1
  38. Rotate(E'_1, If[pi < n ≤ (2 * pi), (-(n - pi)), 6.28], k) → H
  39. CircleArc(F, E', H) → s_1
  40. Rotate(E, If[0 < n ≤ pi, (-n), -3.14], f_1) → E'_2
  41. Rotate(D, If[0 < n ≤ pi, (-n), -3.14], f_1) → D'_2
  42. Mirror(E, f_1) → E'_3
  43. Mirror(D, f_1) → D'_3
  44. CircleArc(I, E, E'_2) → t_1
  45. CircleArc(J, D, D'_2) → c_2
  46. Rotate(D'_3, If[pi < n < (2 * pi), (-(n - pi)), -3.14], f_1) → K
  47. Rotate(E'_3, If[pi < n < (2 * pi), (-(n - pi)), -3.14], f_1) → L
  48. CircleArc(J, D'_3, K) → d_2
  49. CircleArc(I, E'_3, L) → e_2
  50. Rotate(N, If[0 < n ≤ pi, (-n), -3.14], M, k) → N'
  51. Rotate(O, If[0 < n ≤ pi, (-n), -3.14], M, k) → A'
  52. CircleArc(M, N, N') → f_2
  53. CircleArc(M, O, A') → h_2
  54. Mirror(N, k) → N'_1
  55. Mirror(O, k) → O'
  56. Rotate(N'_1, If[pi < n < (2 * pi), (-(n - pi)), -3.14], k) → P
  57. Rotate(O', If[pi < n < (2 * pi), (-(n - pi)), -3.14], k) → O''
  58. CircleArc(M, O', O'') → k_2
  59. CircleArc(M, N'_1, P) → q_2
  60. Rotate(N, If[0 < n ≤ pi, (-n), -3.14], I, f_1) → N'_2
  61. Rotate(O, If[0 < n ≤ pi, (-n), -3.14], I, f_1) → O'_1
  62. CircleArc(Q, N, N'_2) → r_2
  63. CircleArc(R, O, O'_1) → s_2
  64. Mirror(N, f_1) → N'_3
  65. Mirror(O, f_1) → O'_2
  66. Rotate(N'_3, If[pi < n ≤ (2 * pi), (-(n - pi)), -3.14], Q, f_1) → S
  67. Rotate(O'_2, If[pi < n ≤ (2 * pi), (-(n - pi)), -3.14], Q, f_1) → T
  68. CircleArc(Q, N'_3, S) → t_2
  69. CircleArc(R, O'_2, T) → c_3
  70. Textfield(o) → InputBox2
  71. Textfield(a) → InputBox3
  72. Textfield(b) → InputBox4
  73. Textfield(f) → InputBox5
  74. Textfield(h) → InputBox6

AI Drawing Prompt Reference

Draw a surface of revolution formed by rotating the area between two functions, f(x) and g(x). Start by plotting the two function curves, then calculate the integral between them to generate the rotating surface. Also, draw vertical line segments connecting the function curves to the x-axis and rotate them to show the rotation process. Finally, combine the surface with the line segments to form the complete solid of revolution.
The prompt above was compiled from the original drawing commands and is for reference only.