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The Riemann Integral: Mapping Diagram and Graph

The Riemann Integral: Mapping Diagram and Graph
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2018-06-23 06:09
FunctionCalculusIntegral
This example demonstrates the construction of a mapping diagram and graph for the Riemann integral. Key math concepts include the Riemann integral, function graphs, partitioning intervals, and summation approximations. GeoGebra commands/objects used in the construction summary are Sequence, Join, Take, Zip, If, Textfield, Integral, TrapezoidalSum, Sum, Max, Vector, and TableText. It is suitable for high school and above students studying calculus to visually understand the geometric meaning and calculation methods of integrals.

GeoGebra Commands & Parameters

  1. Sequence(1, N) → N1L
  2. Join({{a}, Sort[RandomUniform[a, b, N - 1]], {b}}) → randPart
  3. Take(randPart, 2, N + 1) → randPR
  4. Take(randPart, 1, N) → randPL
  5. Zip(r - l, r, randPR, l, randPL) → h_k
  6. Zip(a + (k * Δx), k, N1L) → ERx_k
  7. If(ER ≟ 2, randPR, ERx_k) → Rx_k
  8. Zip(a + ((k - 1) * Δx), k, N1L) → ELx_k
  9. If(ER ≟ 2, randPL, ELx_k) → Lx_k
  10. Zip(x(Max[f_p, A, B]), A, Lx_k, B, Rx_k) → fmax_k
  11. Sequence(0, N) → N0L
  12. Zip(a + (k * Δx), k, N0L) → EPart
  13. Zip(y(Max[f_p, A, B]), A, Lx_k, B, Rx_k) → fmax_1
  14. Zip(y(Min[f_p, A, B]), A, Lx_k, B, Rx_k) → fmin_1
  15. Zip(RandomUniform[A, B], A, Lx_k, B, Rx_k) → randx_k
  16. If(me ≟ 1, fLx_k, If[me ≟ 2, fRx_k, If[me ≟ 3, fMidx_k, If[me ≟ 4, fmax_1, If[me ≟ 5, fmin_1, franx_k]]]]) → fx_k
  17. If(ER ≟ 1, EfΔx_k, randfΔx_k) → fΔx_k
  18. Textfield(f_p) → inputBox1
  19. If(ER ≟ 1, EPart, randPart) → x_k
  20. Zip((0, t), t, x_k) → x_kMD
  21. Zip((1, Sum[fΔx_k, j]), j, N1L) → Sumpts1
  22. Textfield(a) → inputBox2
  23. Textfield(b) → inputBox3
  24. If(ER ≟ 1, UpperSum[f_p, a, b, N], Sum[(fmax_1 * h_k)]) → UpSum
  25. Integral(f_p, a, b) → Integral
  26. If(ER ≟ 1, LowerSum[f_p, a, b, N], Sum[(fmin_1 * h_k)]) → LowSum
  27. TrapezoidalSum(f_p, a, b, N) → TrapSum
  28. If(ER ≟ 1, Sum[fmidΔx], Sum[(fMidx_k * h_k)]) → MdptSum
  29. Sum(fΔx_k) → IntSum
  30. Zip(x(Min[f_p, A, B]), A, Lx_k, B, Rx_k) → fmin_k
  31. If(me ≟ 1, Lx_k, If[me ≟ 2, Rx_k, If[me ≟ 3, Midx_k, If[me ≟ 4, fmax_k, If[me ≟ 5, fmin_k, randx_k]]]]) → x'_k
  32. Textfield(a) → inputBox2_1
  33. Textfield(b) → inputBox3_1
  34. Zip((0, t), t, x'_k) → c_kMD
  35. Zip(Vector[A, B], A, (0, Rx_k), B, Sumpts1) → Sumarrow
  36. Max(h_k) → mesh
  37. Integral(f_p) → intP
  38. Vector(Element[x_kMD, N + 1], Element[Sumpts1, N]) → f_{arrowb}
  39. Join({{(1, 0)}, Take[Sumpts1, 1, N - 1]}) → Sumpts2
  40. Zip(Vector[A, B], A, Sumpts2, B, Sumpts1) → dy_kMD
  41. Vector(A, Targ0) → u
  42. Zip(Polygon[(s, 0), (s, u), (t, u), (t, 0)], s, Lx_k, t, Rx_k, u, fx_k) → Rects
  43. If(ER ≟ 1, "Euler", "Riemann") → ERText
  44. If(ER ≟ 1, Sum[(fRx_k * Δx)], Sum[(fRx_k * h_k)]) → RtSum
  45. If(ER ≟ 1, LeftSum[f_p, a, b, N], Sum[(fLx_k * h_k)]) → LSum
  46. TableText({Join[{{"c_k"}, x'_k}], Join[{{"P(c_k)"}, fx_k}], Join[{{"P(c_k)Δx"}, fΔx_k}]}, "|_ccv") → TableT
  47. Textfield(N) → InputBox1

AI Drawing Prompt Reference

Draw the Riemann integral graph of a function f(x) over the interval [a, b]. Start by plotting the function and the interval partition lines. Then, calculate the upper and lower sums, and the trapezoidal sum. Next, draw rectangles for each subinterval and connect their tops to form the sum's graphical representation. Finally, show the integral value and the sum approximations.
The prompt above was compiled from the original drawing commands and is for reference only.