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Cylindrical Shells: Forming Solids of Revolution

Cylindrical Shells: Forming Solids of Revolution
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2018-11-20 01:46
SequenceSolid GeometryCalculusIntegralTransformation
In collections GeoGebra 3D & AR: PreCalc & Calculus Resources
This GeoGebra example visually demonstrates the cylindrical shell method for computing volumes of solids of revolution. By rotating function graphs around an axis, it constructs approximating cylindrical shells using polygon rotation and surface dilation, then sums their volumes via definite integration. Core concepts include volume integration, definite integrals, function transformation, rotation geometry, and solid calculus. Construction utilizes GeoGebra commands such as Point, Polygon, Integral, Rotate, Surface, Vector, Dilate, and Sequence to model shell geometry and approximate total volume. Suitable for calculus students and educators seeking an interactive, visual approach to solid volume problems.

GeoGebra Commands & Parameters

  1. Point(x = 0) → A
  2. Point(x = 2) → B
  3. Point(x = 4) → C
  4. Polynomial({A, B, C}) → f
  5. Integral(h, x(A), x(C)) → a
  6. Point(h) → D
  7. Sequence(Polygon[((i * deltax), 0), (((i + 1) * deltax), 0), (((i + 1) * deltax), h(((i + 1) * deltax))), ((i * deltax), h(((i + 1) * deltax)))], i, 0, n - 1, 1) → list1
  8. Polygon(D, (x(D) - deltax, y(D)), (x(D) - deltax, 0), (x(D), 0)) → poly1, d, i, j, k
  9. Line(E, F) → l
  10. Rotate(h, α, l) → h_1
  11. Surface((t * cos(u)), h(x(D)), (t * sin(u)), t, x(D) - deltax, x(D), u, 0, α) → e
  12. Vector(D, (x(D), 0, 0)) → u
  13. Surface((t * cos(u)), 0, (t * sin(u)), t, x(D) - deltax, x(D), u, 0, α) → m
  14. Surface((t * cos(u)), h(t), (t * sin(u)), t, 0, 10, u, 0, α) → b
  15. Rotate(list1, (-α), l) → list1'
  16. Rotate(poly1, (-α), l) → poly1'
  17. Dilate(G, If[0 < w ≤ 1, 1 - w, 0], D) → G'
  18. Segment(G, G') → f_1
  19. Rotate(D, π, l) → D'
  20. Rotate((x(D) - deltax, y(D)), π, l) → J
  21. Dilate(H, If[1 < w ≤ 2, 2 - w, 0], I) → H'
  22. Segment(H, H') → g_1
  23. Dilate(D', If[2 < w ≤ 3, 3 - w, 0], J) → J'
  24. Segment(D', J') → i_1
  25. Distance(D, yAxis) → a_1
  26. Distance(B, yAxis) → e_1
  27. Surface((x(D) * cos(t)), (v * y(D)), (x(D) * sin(t)), t, 0, α, v, 0, 1) → j_1
  28. Surface(((x(D) - deltax) * cos(t)), (v * y(D)), ((x(D) - deltax) * sin(t)), t, 0, α, v, 0, 1) → o
  29. Mirror(D, x = 0) → D'_1
  30. Line(D'_1, K) → p
  31. Mirror(L, D'_1) → L'
  32. Circle(yAxis, D) → q
  33. Circle(yAxis, M) → r
  34. Dilate(D'_1, If[0 < s ≤ 1, 1 - s, 0], L) → P
  35. Dilate(D'_1, If[0 < s ≤ 1, 1 - s, 0], L') → Q
  36. Vector(D'_1, P) → v
  37. Translate(q, v) → q'
  38. Translate(A, v) → A'
  39. Translate(A, Vector[(-v)]) → A'_1
  40. Translate(q, Vector[(-v)]) → q'_1
  41. Translate(D, v) → D'_2
  42. Translate(D, Vector[(-v)]) → D'_3
  43. Line(A', yAxis) → t
  44. Rotate(D'_2, If[0 < s ≤ 1, ((-s) * π), (-π)], A', t) → R
  45. Mirror(R, z = 0) → R'
  46. Translate(r, v) → r'
  47. Translate(r, Vector[(-v)]) → r'_1
  48. Translate(M, v) → M'
  49. Translate(M, Vector[(-v)]) → M'_1
  50. Rotate(M', If[0 < s ≤ 1, ((-s) * π), (-π)], A', t) → M''
  51. Line(A'_1, yAxis) → k_1
  52. Rotate(M'_1, If[0 < s ≤ 1, (s * π), π], A'_1, k_1) → S
  53. CurveCartesian((x(D) * cos(t)), y(D), (x(D) * sin(t)), t, 0, α) → c_1
  54. CurveCartesian((x(M) * cos(t)), y(M), (x(M) * sin(t)), t, 0, α) → l_1
  55. Rotate(D, (-α), yAxis) → D'_4
  56. Rotate(M, (-α), yAxis) → M'_2
  57. Segment(D'_4, M'_2) → m_1
  58. CurveCartesian((x(D) * cos(t)), 0, (x(D) * sin(t)), t, 0, α) → n_1
  59. CurveCartesian((x(M) * cos(t)), 0, (x(M) * sin(t)), t, 0, α) → o_1
  60. Polygon(T, U, M'', R) → q1, t_1, u_1, m'', r_1
  61. Polygon(S, R', W, V) → q2, s_1, r'_2, w_1, v_1
  62. Mirror(M, x = 0) → M'_3
  63. Translate(M'_3, v) → Z
  64. Translate(M'_3, Vector[(-v)]) → A_1
  65. Center(r') → B_1
  66. CircleArc(B_1, P, R) → q_1
  67. CircleArc(B_1, Z, M'') → c_2
  68. Mirror(q_1, z = 0) → d_2
  69. Mirror(c_2, z = 0) → e_2
  70. Angle(Z, B_1, M'') → β
  71. Surface(x(B_1) - (x(D) * cos(t)), (v * y(D)), z(B_1) + (x(D) * sin(t)), t, 0, β, v, 0, 1) → a_2
  72. Surface(x(B_1) - (x(M) * cos(t)), (v * y(D)), z(B_1) + (x(M) * sin(t)), t, 0, β, v, 0, 1) → b_2
  73. Surface(x(B_1) + (t * cos(u)), y(D), z(B_1) - (t * sin(u)), t, x(P), x(P) + deltax, u, 0, β) → i_2
  74. Surface(x(B_1) + (t * cos(u)), 0, z(B_1) - (t * sin(u)), t, x(P), x(P) + deltax, u, 0, β) → j_2
  75. Mirror(a_2, z = 0) → k_2
  76. Mirror(b_2, z = 0) → l_2
  77. Mirror(i_2, z = 0) → m_2
  78. Mirror(j_2, z = 0) → n_2
  79. Segment(R, M'') → p_2
  80. Segment(M'', U) → q_2
  81. Segment(U, T) → r_2
  82. Segment(T, R) → s_2
  83. Vector(D, (x(D), 0, 0)) → u_2
  84. Translate(q_1, u_2) → t_2
  85. Translate(c_2, u_2) → c_3
  86. Segment(S, R') → f_3
  87. Segment(R', W) → g_3
  88. Segment(W, V) → h_3
  89. Segment(V, S) → i_3
  90. Translate(e_2, u_2) → d_3
  91. Translate(d_2, u_2) → e_3
  92. Mirror(E_1, z = 0) → G_1
  93. Mirror(F_1, z = 0) → H_1
  94. Polygon(Q, P, E_1, G_1) → q3, q_3, p_3, e_4, g_4
  95. Polygon(A_1, H_1, F_1, Z) → q4, a_3, h_4, f_4, z_1
  96. Segment(E_1, G_1) → l_3
  97. Segment(F_1, H_1) → j_3
  98. Segment(P, Q) → p_1
  99. Segment(A_1, Z) → f_2
  100. Polygon(Q, A_1, Z, P) → q5, q_4, a_4, z_2, p_4
  101. Polygon(E_1, G_1, H_1, F_1) → q6, e_5, g_2, h_2, f_5
  102. Translate(D'_4, u_2) → I_1
  103. Translate(M'_2, u_2) → J_1
  104. Segment(I_1, J_1) → k_3
  105. Distance(L', L) → distanceL'L