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圆柱壳法求固体体积

圆柱壳法求固体体积
几何画板及绘图内容加载中，请稍等……
2018-11-20 01:46
数列立体几何微积分积分变换
所属专题 GeoGebra 3D & AR: PreCalc & Calculus Resources
演示如何使用圆柱壳法通过围绕坐标轴旋转函数图像来生成实体。涉及的核心数学概念包括：圆柱壳体积积分、定积分、函数变换、旋转几何以及微积分在求体积问题中的应用。作图要点提炼：使用了 Point、Polygon、Integral、Rotate、Surface、Vector、Dilate、Sequence 等 GeoGebra 命令，通过函数图像的旋转与拼接构建圆柱壳模型，并利用 Definite Integral 与 Sequence 实现多壳体积分逼近。适合高中微积分或大学一年级数学学习，特别是求解固体体积的几何直观演示。

GeoGebra 命令及参数

  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