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The 3-Level Formula for Volumes Calculus

The 3-Level Formula for Volumes Calculus
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2020-06-22 03:28
FunctionSolid GeometryCalculusIntegralGeometric Construction
This GeoGebra example dynamically illustrates the three-level formula for calculating volumes of cones, cylinders, and spheres, showing how base area, cross-section, and height combine into volume integrals. Core mathematical concepts include integral calculus, solids of revolution, and volume relationships among basic 3D shapes. Construction highlights: uses Ellipse and Circle to define bases and cross-sections, Center and Midpoint for symmetry, Line, Segment, and Intersect for auxiliary constructions, Translate and Point for 3D projection, and multiple text objects (texte1, texte2, texte3) to annotate formulas and values. Suitable for high school to early college students learning geometric volume derivation, and for introductory calculus classes demonstrating the connection between definite integrals and volume.

GeoGebra Commands & Parameters

  1. Ellipse(A, B, C) → c
  2. Ellipse(D, E, F) → d
  3. Ellipse(G, H, I) → e
  4. Center(c) → J
  5. Center(d) → K
  6. Center(e) → L
  7. Line(K, yAxis) → g
  8. Line(L, g) → h
  9. Point(g) → N
  10. Point(h) → O
  11. Translate(d, Vector[K, N]) → d'
  12. Line(D, E) → i
  13. Line(G, H) → j
  14. Intersect(d, i) → P, Q
  15. Intersect(e, i) → R, S
  16. Segment(O, G) → k
  17. Segment(O, H) → l
  18. Segment(N, K) → m
  19. Line(J, yAxis) → f
  20. Point(f) → M
  21. Circle(J, M) → p
  22. Intersect(p, f) → T, U
  23. Segment(O, L) → n
  24. Line(Q, m) → r
  25. Line(P, m) → q
  26. Intersect(d', q) → V, W
  27. Intersect(d', r) → Z, A_1
  28. Midpoint(N, K) → B_1
  29. Midpoint(L, O) → C_1
  30. Segment(M, U) → s
  31. Segment(V, P) → t
  32. Segment(Z, Q) → a
  33. Point(l) → D_1
  34. Point(k) → E_1
  35. Point(n) → F_1
  36. Ellipse(D_1, E_1, F_1) → c_1
  37. Translate(d, Vector[K, B_1]) → d'_1
  38. Ellipse(G_1, H_1, I_1) → d_1
  39. Center(d_1) → J_1
  40. Line(J_1, n) → b
  41. Point(b) → K_1
  42. Ellipse(L_1, M_1, K_1) → e_1
  43. Segment(L_1, G_1) → f_1
  44. Segment(M_1, H_1) → g_1
  45. Center(e_1) → N_1
  46. Midpoint(N_1, J_1) → O_1
  47. Segment(N_1, J_1) → h_1
  48. Ellipse(P_1, Q_1, R_1) → k_1
  49. Segment(M, J) → i_1
  50. Segment(J, U) → j_1
  51. Segment(N, B_1) → l_1
  52. Segment(B_1, K) → m_1
  53. Segment(O, C_1) → n_1
  54. Segment(C_1, L) → p_1
  55. Segment(N_1, O_1) → q_1
  56. Segment(O_1, J_1) → r_1