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Proof of Sum Property for the Definite Integral

Proof of Sum Property for the Definite Integral
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2018-07-12 05:12
FunctionCalculusIntegral
This example demonstrates the proof of the sum property for the definite integral. Key mathematical concepts include definite integral and calculus. Using GeoGebra commands such as Integral, Textfield, Point, etc., the function graph and key points are constructed to illustrate the properties of definite integrals. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

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