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积分作为f(x)值的累积：动态可视化

积分作为f(x)值的累积：动态可视化
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2016-06-20 20:54
多边形函数微积分积分滑块动画
① 本范例通过动态图形直观展示定积分作为函数f(x)值在区间上的累积过程，即面积累加形成积分值。② 核心数学概念是定积分的几何意义——曲线下面积，以及黎曼和的思想，涉及函数、区间分割、矩形面积求和。③ 作图要点：使用GeoGebra命令如Point、OrthogonalLine、Intersect、Segment、Polygon、Vector、Translate、Ray等构造点、垂线、交点、线段、多边形和向量，通过平移和射线生成多个矩形条，对象包括A_0至A_6、Y_0至Y_5等，体现分割与累积。④ 适合高中至大学微积分入门教学，帮助学生理解积分与面积的关系，也可用于探究黎曼和逼近。

GeoGebra 命令及参数

  1. Point({0, 0}) → A_0
  2. Point({1, 0}) → A_1
  3. Point({2, 0}) → A_2
  4. Point({3, 0}) → A_3
  5. Point({4, 0}) → A_4
  6. Point({5, 0}) → A_5
  7. OrthogonalLine(A_0, xAxis) → u
  8. OrthogonalLine(A_1, xAxis) → g
  9. OrthogonalLine(A_2, xAxis) → h
  10. OrthogonalLine(A_3, xAxis) → i
  11. OrthogonalLine(A_4, xAxis) → j
  12. OrthogonalLine(A_5, xAxis) → k
  13. Point(u) → Y_0
  14. Point(g) → Y_1
  15. Point(h) → Y_2
  16. Point(i) → Y_3
  17. Point(j) → Y_4
  18. Point(k) → Y_5
  19. OrthogonalLine(Y_0, u) → l
  20. OrthogonalLine(Y_1, g) → m
  21. OrthogonalLine(Y_2, h) → n
  22. OrthogonalLine(Y_3, i) → p
  23. OrthogonalLine(Y_4, j) → q
  24. OrthogonalLine(Y_5, k) → r
  25. Intersect(g, l) → A
  26. Intersect(h, m) → B
  27. Intersect(i, n) → C
  28. Intersect(j, p) → D
  29. Intersect(k, q) → E
  30. Point({6, 0}) → A_6
  31. OrthogonalLine(A_6, xAxis) → s
  32. Intersect(r, s) → F
  33. Segment(Y_0, A) → t
  34. Segment(Y_1, B) → b
  35. Segment(Y_2, C) → d
  36. Segment(Y_3, D) → f_1
  37. Segment(Y_4, E) → i_1
  38. Segment(Y_5, F) → k_1
  39. Segment(A_6, A_5) → a
  40. Segment(A_4, A_3) → c
  41. Segment(A_2, A_1) → e
  42. Polygon(Y_0, A, A_1, A_0) → многоугольник1, y_0, a_1, a_2, a_0
  43. Polygon(Y_1, B, A_2, A_1) → многоугольник2, y_1, b_1, a_3, a_4
  44. Polygon(Y_2, C, A_3, A_2) → многоугольник3, y_2, c_1, a_5, a_6
  45. Polygon(Y_3, D, A_4, A_3) → многоугольник4, y_3, d_1, a_7, a_8
  46. Polygon(Y_4, E, A_5, A_4) → многоугольник5, y_4, e_1, a_9, a_{10}
  47. Polygon(Y_5, F, A_6, A_5) → многоугольник6, y_5, f_2, a_{11}, a_{12}
  48. Vector(A_0, Y_0) → v
  49. Vector(A_1, Y_1) → w
  50. Vector(A_2, Y_2) → u_1
  51. Vector(A_3, Y_3) → v_1
  52. Vector(A_4, Y_4) → w_1
  53. Vector(A_5, Y_5) → g_1
  54. Vector(A_0, A_1) → j_1
  55. Translate(A, j_1) → A'
  56. Translate(A', w) → A''
  57. Vector(A', A'') → l_1
  58. Translate(A'', j_1) → A'''
  59. Vector(A'', A''') → m_1
  60. Translate(A''', u_1) → G
  61. Vector(A''', G) → n_1
  62. Translate(G, j_1) → G'
  63. Vector(G, G') → p_1
  64. Translate(G', v_1) → G''
  65. Vector(G', G'') → q_1
  66. Translate(G'', j_1) → G'''
  67. Vector(G'', G''') → r_1
  68. Translate(G''', w_1) → H
  69. Vector(G''', H) → s_1
  70. Translate(H, j_1) → H'
  71. Vector(H, H') → t_1
  72. Translate(H', g_1) → H''
  73. Vector(H', H'') → u_2
  74. Translate(A_1, v) → I
  75. Vector(A_1, I) → v_2
  76. Vector(I, A') → h_1
  77. Ray(I, A') → g_2
  78. Ray(A'', A''') → h_2
  79. Ray(G, G') → i_2
  80. Ray(G'', G''') → j_2
  81. Ray(H, H') → k_2
  82. Intersect(u, xAxis) → J
  83. Segment(J, I) → l_2
  84. Segment(I, A'') → m_2
  85. Segment(A'', G) → n_2
  86. Segment(G, G'') → p_2
  87. Segment(G'', H) → q_2
  88. Segment(H, H'') → r_2
  89. Segment(A, Y_1) → f
  90. Segment(B, Y_2) → s_2
  91. Segment(Y_3, C) → t_2
  92. Segment(D, Y_4) → b_2
  93. Segment(Y_5, E) → c_2