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Demonstration of Upper and Lower Sums/Riemann Sum

Demonstration of Upper and Lower Sums/Riemann Sum
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2021-12-15 03:40
CalculusInequalityGeometric Construction
In collections Open Middle: Distance and Midpoint Exercise
This example demonstrates how to dynamically construct upper and lower sums to approximate the Riemann sum, visually illustrating the geometric meaning of the integral concept. It involves core mathematical concepts such as upper sum, lower sum, Riemann sum, similar triangles, inequalities, and arithmetic mean. The construction uses GeoGebra commands and objects like PointIn, Segment, SurdText, Textfield, Point, Mode, Intersect, Polygon, Midpoint, If, Angle. It is suitable for high school calculus introduction, helping students understand the discrete approximation process of integration.

GeoGebra Commands & Parameters

  1. PointIn(e) → A
  2. PointIn(e) → B
  3. Segment(A, B) → f
  4. SurdText(f) → text1
  5. Textfield(j) → InputBox1
  6. Textfield(k) → InputBox2
  7. Point(Circle[A, sqrt((100 * j) + (10 * k) + o)]) → C
  8. Segment(A, C) → l
  9. Mode(l1) → l2
  10. Intersect(eq1, p) → D
  11. Polygon(A, D, M) → t1, f_1, g_1, h_1
  12. Segment(A, D) → q
  13. Segment(D, B) → r
  14. Textfield(o) → InputBox3
  15. Midpoint(Segment[A, B]) → E
  16. If(mcorrect, "√", "x") → text9
  17. Segment(A, M) → i_1
  18. Segment(D, M) → j_1
  19. Angle(M, D, A) → α
  20. Segment(A, D) → k_1
  21. Segment(A, M) → l_1
  22. Segment(M, B) → m_1
  23. If(Distance[A, C] < Distance[A, M], "Too short!", Distance[A, C] > Distance[A, M], "Too long!", "This value is correct!") → text7
  24. If(Length[l2] ≟ 0, "All digits are different.", "Some digits repeat.") → text8

AI Drawing Prompt Reference

Create a dynamic geometry illustration of upper and lower sums to approximate a Riemann sum. Start with two fixed interval endpoints and draw the segment between them. From the first endpoint, generate a third point on a circle whose radius is determined by three user-controlled numeric inputs; connect this point to the first endpoint to form the main boundary lines. Construct additional polygonal regions that represent the upper and lower sum areas, including lines passing through the interval midpoint. Add a marked angle at the midpoint with respect to the endpoint rays. Include on-canvas text feedback that reacts to whether a chosen length is too short/too long, and another text indicating whether the supplied digits satisfy the stated condition. The figure should update as the inputs change to visualize the inequality-based comparison of sums.
The prompt above was compiled from the original drawing commands and is for reference only.