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Volume of the Sphere

Volume of the Sphere
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2020-01-06 16:52
Solid GeometryCalculusIntegral
This example demonstrates the calculation of a sphere's volume. Key mathematical concepts include spheres, volume, and integration. Using GeoGebra commands such as Intersect, Circle, and Sequence, the process of calculating the volume of a sphere is dynamically illustrated. Suitable for high school and above students studying solid geometry and calculus.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Circle(A, 1) → c
  3. Sequence((0, i / dx), i, 0, dx) → list1
  4. Sequence((sqrt(1 - (i / dx)^(2)), i / dx), i, 0, dx) → list2
  5. Point(list1) → B
  6. Point(list2) → C
  7. Intersect(c, xAxis) → E, F
  8. Point(list2) → H
  9. Point(list1) → I
  10. Point(list2) → J
  11. Point(list2) → K
  12. Point(list1) → L
  13. Point(list1) → M
  14. Segment(M, L) → q
  15. Point(list1) → R
  16. Point(list2) → S
  17. Point(list1) → T
  18. Point(list2) → U
  19. Sequence(((-sqrt(1 - (i / dx)^(2))), i / dx), i, 0, dx) → list3
  20. Point(list3) → D
  21. Point(list2) → G
  22. Segment(D, G) → f
  23. Point(list2) → W
  24. Point(list3) → V
  25. Segment(V, W) → g
  26. Point(list2) → A_1
  27. Point(list3) → Z
  28. Point(list3) → B_1
  29. Point(list3) → C_1
  30. Point(list3) → D_1
  31. Segment(Z, A_1) → h
  32. Point(list2) → E_1
  33. Segment(B_1, E_1) → i
  34. Point(list2) → F_1
  35. Segment(C_1, F_1) → j
  36. Point(list2) → G_1
  37. Segment(D_1, G_1) → r
  38. Point(list3) → H_1
  39. Point(list3) → I_1
  40. Segment(H_1, K) → k
  41. Segment(J, I_1) → l
  42. Intersect(c, j, 1) → N
  43. Intersect(c, i, 1) → O
  44. Intersect(c, i, 2) → P
  45. Intersect(c, j, 2) → Q
  46. Point(list3) → J_1
  47. Point(list2) → K_1
  48. Point(list2) → L_1
  49. Point(list3) → M_1
  50. Point(list3) → N_1
  51. Point(list2) → O_1
  52. Segment(J_1, K_1) → m
  53. Segment(L_1, M_1) → s
  54. Segment(N_1, O_1) → t
  55. Segment(A, Q) → a
  56. OrthogonalLine(Q, xAxis) → b
  57. OrthogonalLine(N, xAxis) → d
  58. Intersect(b, i) → P_1
  59. Intersect(d, i) → Q_1
  60. Polygon(N, Q_1, P_1, Q) → poly1, n_1, q_1, p_1, q_2
  61. Point(list1) → R_1
  62. Sequence(Segment[(sqrt(1 - (i / dx)^(2)), i / dx), ((-sqrt(1 - (i / dx)^(2))), i / dx)], i, 1, dx) → list4
  63. Sequence((0, 0, i / dx), i, 0, dx) → l1
  64. Sequence(((-sqrt(1 - (i / (dx + 1))^(2))), 0, i / (dx + 1)), i, 0, dx + 1) → l3
  65. Sequence((sqrt(1 - (i / (dx + 1))^(2)), 0, i / (dx + 1)), i, 0, dx + 1) → l2
  66. Sequence((0, sqrt(1 - (i / (dx + 1))^(2)), i / (dx + 1)), i, 0, dx + 1) → l4
  67. Sequence(Circle[Element[l2, i], Element[l3, i], Element[l4, i]], i, 1, Length[l2]) → l5
  68. Distance(Element[l1, 2], Element[l1, 3]) → hh
  69. Sequence(Cylinder[Element[l5, i], hh], i, 1, Length[l5]) → l6
  70. Element(l6, 5) → e
  71. Sphere(A, 1) → o
  72. Midpoint(Q, A) → S_1
  73. Sequence((x(Element[list3, i]), y(Element[list3, i - 1])), i, 1, Length[list3]) → list3b
  74. Sequence((x(Element[list2, i]), y(Element[list2, i - 1])), i, 1, Length[list2]) → list2b
  75. Sequence(Polygon[Element[list2, i], Element[list2b, i], Element[list3b, i], Element[list3, i]], i, 1, Length[list2]) → l7
  76. Sequence(CurveCartesian[x(A) + (cos(i) * cos(j)), y(A) + (cos(i) * sin(j)), z(A) + sin(i), i, 0, (2 * π)], j, 0, (2 * π), (2 * π / 12)) → l8
  77. Sequence(CurveCartesian[x(A) + (cos(i) * cos(j)), y(A) + (cos(i) * sin(j)), z(A) + sin(i), j, 0, (2 * π)], i, 0, (2 * π), π / 12) → l9
  78. Midpoint(N, Q) → T_1
  79. Segment(T_1, A) → n
  80. Mirror(l6, A) → l6'
  81. Mirror(l7, A) → l7'