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GeoGebra Demonstration of Additive Property of Definite Integral

GeoGebra Demonstration of Additive Property of Definite Integral
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2018-07-14 07:39
FunctionCalculusIntegral
This example dynamically demonstrates the additive property of definite integrals using GeoGebra, showing that the integral over a combined interval equals the sum of integrals over subintervals. Core math concepts include definite integrals, function graphs, interval division, and vector operations. Construction highlights involve commands like Sequence, Integral, Point, Line, Vector, etc., as per the summary to build function graphs, segment lines, and compute integrals. Suitable for high school and above to understand the geometric meaning of the additive property theorem.

GeoGebra Commands & Parameters

  1. Sequence(1, n) → N1
  2. Sequence(0, n) → N0
  3. Zip(a - dx / 2 + (k * dx), k, N1) → c_*
  4. Zip(a + (k * dx), k, N0) → x_*
  5. IndexOf(1, Findc) → c_#
  6. Integral(P, a, c) → IPac
  7. Integral(P, c, b) → IPcb
  8. Integral(P, a, b) → IPab
  9. Textfield(P) → inputBox1
  10. Point(yAxis) → D
  11. Point(yAxis) → E
  12. Line(D, E) → Xaxis
  13. Line(Ypt, Xaxis) → Yaxis
  14. Vector(G, J) → u
  15. Vector(I, F) → v
  16. Vector(H, K) → w
  17. Segment(F, K) → f
  18. Segment(F, J) → h
  19. Segment(I, H) → i
  20. Segment(G, I) → j
  21. Integral(αP, a, b) → IαPab
  22. Line(M, Yaxis) → Y2axis
  23. Vector(J, αs_a) → e
  24. Vector(K, αs_b) → d
  25. Segment(αs_a, αs_b) → g
  26. Segment(J, K) → m
  27. Segment(G, H) → p
  28. Textfield(Q) → InputBox1
  29. Integral(Q, a, b) → IQab
  30. Integral(P(x) + Q(x), a, b) → IPQab
  31. Line(O, Xaxis) → q
  32. Vector(G, SQa) → r
  33. Vector(H, SQb) → s
  34. Segment(SQb, SQa) → t
  35. Vector(K, SPQb) → u_1
  36. Vector(SQb, SPQb) → v_1
  37. Vector(SQa, SPQa) → w_1
  38. Segment(SPQb, SPQa) → f_1
  39. Vector(SQb, SQb1) → a_1
  40. Segment(SPQb, SQb1) → g_1
  41. Segment(SQb1, SPQa) → h_1
  42. Vector(J, SQb1) → b_1
  43. Point(f_1) → R
  44. Zip((0, a + (k * dx)), k, N0) → x_1
  45. Zip(Sum[Pcdx_*, k], k, N0) → SPcdx_*
  46. Zip(Sum[αPxdx_*, k], k, N0) → SαPcdx_*
  47. Zip((x(Ypt), sp), sp, SPcdx_*) → SptPcdx_*
  48. Zip(Vector[A, B], A, x_1, B, SptPcdx_*) → vecSP_*
  49. Zip((x(M), sp), sp, SαPcdx_*) → SptαPcdx_*
  50. Zip(Vector[A, B], A, SptPcdx_*, B, SptαPcdx_*) → vecSαP_*
  51. TableText(Transpose[{Join[{{"c_k"}, c_*}], Join[{{"P(c_k)Δx"}, Pcdx_*}], Join[{{"αP(c_k)Δx"}, αPxdx_*}]}], "|_cc") → TableI
  52. RectangleSum(P, y(G), y(H), n, 1 / 2) → GrMidEstIntP
  53. RectangleSum((α * P(x)), y(G), y(H), n, 1 / 2) → GrMidestIntαP
  54. Zip(Sum[Qcdx_*, k], k, N0) → SQcdx_*
  55. Zip((x(M), sp), sp, SQcdx_*) → SptQcdx_*
  56. Zip(Vector[A, B], A, x_1, B, SptQcdx_*) → vecSQ_*
  57. Zip(Sum[PQcdx_*, k], k, N0) → SPQcdx_*
  58. Zip((x(O), sp), sp, SPQcdx_*) → SptPQcdx_*
  59. Zip(Vector[A, B], A, SptPcdx_*, B, SptPQcdx_*) → vecSP2PQ_*
  60. Zip(Vector[A, B], A, SptQcdx_*, B, SptPQcdx_*) → vecSQ2PQ_*
  61. RectangleSum(Q, y(G), y(H), n, 1 / 2) → GrMidEstIntQ
  62. RectangleSum(PQ, y(G), y(H), n, 1 / 2) → GrMidEstIntPQ
  63. Zip((x(O), sp), sp, SQcdx_*) → SptQ'cdx_*
  64. Zip(Vector[A, B], A, SptQcdx_*, B, SptQ'cdx_*) → vecSQ'_*
  65. Zip((x(O), sp), sp, SPcdx_* + GrMidEstIntQ) → SptP'cdx_*
  66. Zip(Vector[A, B], A, SptPcdx_*, B, SptP'cdx_*) → vecSP'_*
  67. Vector(G, M_0) → d_1
  68. Vector(G, O_0) → e_1
  69. TableText(Transpose[{Join[{{"c_k"}, c_*}], Join[{{"P(c_k)Δx"}, Pcdx_*}]}], "|_cc") → TableP
  70. Element(x_*, c_# - 1) → c1
  71. RectangleSum(P, y(G), c1, c_# - 2, 1 / 2) → GrMidEstIntP1
  72. Element(x_*, c_#) → c2
  73. RectangleSum(P, c2, y(H), n - c_# + 1, 1 / 2) → GrMidEstIntP2
  74. First(c_*, c_# - 2) → c1_*
  75. Last(c_*, n - c_# + 1) → c2_*
  76. Join({c1_*, {c}, c2_*}) → ca_*
  77. Zip(Sum[Pcadx_*, k], k, N0) → SPcadx_*
  78. Zip((x(Ypt), sp), sp, SPcadx_*) → SptPcadx_*
  79. Zip(Vector[A, B], A, x_1, B, SptPcadx_*) → vecSPa_*

AI Drawing Prompt Reference

Draw a function graph to demonstrate the additive property of definite integrals. Mark three intervals [a, c], [c, b], and [a, b] on the x-axis, and calculate and display the definite integral values for each interval. Use geometric shapes and vector operations to visually show that the integral over [a, b] equals the sum of the integrals over [a, c] and [c, b]. Start by plotting the function curve and coordinate axes, then label key points and lines, and finally use vectors to represent the integral regions to complete the construction.
The prompt above was compiled from the original drawing commands and is for reference only.