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The definite integral as the limit of a Riemann sum

The definite integral as the limit of a Riemann sum
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2017-04-11 23:54
FunctionCalculusIntegralLimitGeometric Construction
This example demonstrates how the definite integral can be calculated as the limit of a Riemann sum, visually illustrating the relationship between integration and area. Core mathematical concepts include definite integral, Riemann sum, trapezoidal rule, and vectors. GeoGebra commands and objects like If, Sequence, Integral, LaTeX, Segment, FractionText, Midpoint, Vector, Distance, Intersect, and Point are used to construct the animation showing the integration process. Suitable for high school and above students to learn and understand basic calculus concepts.

GeoGebra Commands & Parameters

  1. If(f_{alt}, f_2, f) → f_1
  2. Sequence(a + (i * (b - a) / n), i, 0, n) → x_i
  3. Integral(f_1, a, b) → S_{actual}
  4. Sequence((1 / 2 * (a + (i * (b - a) / n) + a + ((i + 1) * (b - a) / n))), i, 0, n) → mid
  5. Sequence(f_1((1 / 2 * (a + (i * (b - a) / n) + a + ((i + 1) * (b - a) / n)))), i, 0, n - 1) → functionvaluesmid
  6. Sequence(Polygon[(Element[x_i, i], 0), (Element[x_i, i + 1], 0), (Element[x_i, i + 1], Element[functionvaluesmid, i]), (Element[x_i, i], Element[functionvaluesmid, i])], i, 1, n) → poly_{mid}
  7. Sequence(f_1((p_1 * (a + (i * (b - a) / n))) + ((1 - p_1) * (a + ((i + 1) * (b - a) / n)))), i, 0, n - 1) → functionvaluesp
  8. Sequence(Polygon[(Element[x_i, i], 0), (Element[x_i, i + 1], 0), (Element[x_i, i + 1], Element[functionvaluesp, i]), (Element[x_i, i], Element[functionvaluesp, i])], i, 1, n) → poly4
  9. LaTeX("Midpoints: " + S_{mid}) → text4
  10. Sequence((Element[mid, i], Element[functionvaluesmid, i]), i, 1, n) → pointsmid
  11. Sequence(((p_1 * (a + (i * (b - a) / n))) + ((1 - p_1) * (a + ((i + 1) * (b - a) / n))), f_1((p_1 * (a + (i * (b - a) / n))) + ((1 - p_1) * (a + ((i + 1) * (b - a) / n))))), i, 0, n - 1) → pointsp
  12. LaTeX("p=" + p_1 + ",\;(1-p)=" + (1 - p_1) + "\;Center\;Points:" + S_p) → text5
  13. Segment(A_1, B_2) → l
  14. LaTeX("f(x)\;=\;" + f_1) → text9
  15. Sequence(Polygon[(Element[x_i, i], 0), (Element[x_i, i + 1], 0), (Element[x_i, i + 1], Element[y_i, i]), (Element[x_i, i], Element[y_i, i])], i, 1, n) → poly_{left}
  16. FractionText(b) → text_b
  17. FractionText(a) → text_a
  18. Sequence((Element[x_i, i], Element[y_i, i]), i, 1, n) → pointsleft
  19. Sequence((Element[x_i, i + 1], Element[y_i, i + 1]), i, 1, n) → pointsright
  20. Sequence(Polygon[(Element[x_i, i], 0), (Element[x_i, i + 1], 0), (Element[x_i, i + 1], Element[y_i, i + 1]), (Element[x_i, i], Element[y_i, i + 1])], i, 1, n) → poly_{right}
  21. Sequence(Polygon[(Element[x_i, i], 0), (Element[x_i, i + 1], 0), (Element[x_i, i + 1], Element[y_i, i + 1]), (Element[x_i, i], Element[y_i, i])], i, 1, n) → poly_{trap}
  22. Sequence((Element[x_i, i], Element[y_i, i]), i, 1, n + 1) → pointstrap
  23. LaTeX("Right\;Endpoints: " + S_{right}) → text1_1
  24. LaTeX("Left\;Endpoints: " + S_{left}) → text3
  25. Midpoint(H, J) → E
  26. Vector(E, J) → u
  27. Vector(E, H) → v
  28. Distance(H, J) → distanceHJ
  29. Intersect(F, r, (2.75, 26.46875)) → G
  30. Intersect(F, s, (0.75, 4.21875)) → I
  31. Segment(B_2, G) → m
  32. Point(poly_{left}) → K
  33. Segment(K, I) → t
  34. Integral(f_1) → F_0
  35. LaTeX("F(x)\;=\int f(x) \, dx=\;" + F_0 + " + C") → text_F
  36. Segment(D, G) → a_1
  37. Segment(C_1, I) → b_1
  38. Intersect(f_1, s, (0.75, 10.125)) → A
  39. Intersect(f_1, r, (2.75, 4.125)) → B
  40. Segment(A, A_1) → h
  41. Segment(B, B_2) → j
  42. LaTeX("Trapezoid\;Rule: " + S_{trap}) → text6