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Riemann Sum Model

Riemann Sum Model
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2015-10-12 02:11
SequenceIntegralProbability & Statistics
This model demonstrates the concept of Riemann sums, approximating the integral value by dynamically adjusting the number of segments. Key mathematical concepts include integration and stochastic processes. It uses commands like point, sequence, sort, sum, and integral in GeoGebra. Suitable for calculus education to help students understand the geometric meaning of integration.

GeoGebra Commands & Parameters

  1. Point(xAxis) → B
  2. Point(xAxis) → A
  3. Sequence(RandomUniform[a, b], i, 1, N - 1) → rands
  4. Sort(rands) → sortrands
  5. Sum(rands) → sumrands
  6. Textfield(f_1) → inputBox1
  7. Append(Append[A, Sequence[(Element[sortrands, i], 0), i, 1, N - 1]], B) → partitionP
  8. Sequence(f(RandomUniform[x(Element[partitionP, i]), x(Element[partitionP, i + 1])]), i, 1, N) → fx_i
  9. Sequence((0, Element[fx_i, i]), i, 1, N) → fx_ipointlist
  10. Sequence(Polygon[Element[partitionP, i], Element[partitionP, i] + Element[fx_ipointlist, i], Element[partitionP, i + 1] + Element[fx_ipointlist, i], Element[partitionP, i + 1]], i, 1, N) → polygons
  11. Integral(f, a, b) → integral
  12. Sequence(If[Element[fx_i, i] > 0, Area[Element[polygons, i]], (-Area[Element[polygons, i]])], i, 1, N) → signedpolygons
  13. Sum(signedpolygons) → integralapprox
  14. Max(Sequence[x(Element[partitionP, i + 1]) - x(Element[partitionP, i]), i, 1, N]) → normP

AI Drawing Prompt Reference

Draw a model demonstrating the concept of Riemann sums, featuring two points on the x-axis, randomly generating N-1 numbers between a and b, sorting them, creating segments on the x-axis, and placing a rectangle on each segment with a random height. Sum the areas of all rectangles to approximate the integral, display the true integral value, and show the width of the segments to illustrate the geometric meaning of integration.
The prompt above was compiled from the original drawing commands and is for reference only.