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Integral as Accumulation of f(x) Values: Dynamic Visualization

Integral as Accumulation of f(x) Values: Dynamic Visualization
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2016-06-20 20:54
PolygonFunctionCalculusIntegralSlider Animation
① This example dynamically visualizes the definite integral as the accumulation of function values f(x) over an interval, showing how areas sum to the integral. ② Core mathematical concepts include the geometric interpretation of definite integrals (area under a curve) and Riemann sums, involving functions, interval partitioning, and summation of rectangular areas. ③ Construction details: GeoGebra commands such as Point, OrthogonalLine, Intersect, Segment, Polygon, Vector, Translate, and Ray are used to create points, perpendicular lines, intersections, segments, polygons, and vectors, with translations and rays generating multiple rectangular strips; objects include A_0 to A_6, Y_0 to Y_5, etc., illustrating partitioning and accumulation. ④ Suitable for high school to college introductory calculus, helping students understand the relationship between integrals and area, and explore Riemann sum approximations.

GeoGebra Commands & Parameters

  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