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Definite Integral Illustrator (I) by HUMARAN MARTINEZ, YUITZA

Definite Integral Illustrator (I) by HUMARAN MARTINEZ, YUITZA
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2019-11-27 21:47
PolygonFunctionCalculusIntegralSlider Animation
① This example dynamically illustrates the geometric meaning of a definite integral as the area under a function curve, approximated by left and right Riemann sums. ② Core mathematical concepts include definite integrals, Riemann sums, limits, function graphing, interval partitioning, and area computation. ③ Construction highlights: uses GeoGebra commands such as Textfield for input boxes, Sequence to generate partition points, Translate to shift rectangles, LeftSum and Sum to compute left/right sums, Intersect for intersection points, Polygon to build rectangles, and Integral to render the exact area; key objects include function f, endpoints a and b, number of subintervals n, rectangle heights, midpoints, left/right endpoints, and rectangle sums. ④ Suitable for high school to early college calculus courses, helping students grasp the definition of the definite integral, the Riemann sum approximation process, and the relationship between integrals and area, as well as exploring how the number of partitions n affects approximation accuracy.

GeoGebra Commands & Parameters

  1. Textfield(f) → InputBox1
  2. Textfield(a) → InputBox2
  3. Textfield(b) → InputBox3
  4. Sequence((a + (i * deltax), 0), i, 1, n, 1) → rightx
  5. Sequence((a + (i * deltax), f(a + (i * deltax))), i, 1, n, 1) → fofrightx
  6. Sequence((a + (i * deltax), 0), i, 0, n - 1, 1) → leftx
  7. Sequence(Midpoint[Element[leftx, i], Element[rightx, i]], i, 1, n, 1) → midpoints
  8. Sequence((a + (i * deltax), f(a + (i * deltax))), i, 1, n, 1) → list1
  9. Translate(list1, (deltax, 0)) → list1'
  10. LeftSum(f, a, b, n) → l
  11. Sequence(x = x(Element[midpoints, i]), i, 1, n, 1) → list2
  12. Sequence(Intersect[f, Element[list2, i], (3, 6.079441541679836)], i, 1, n, 1) → list3
  13. Translate(list3, (deltax / 2, 0)) → midpointtopright
  14. Translate(list3, (((-deltax)) / 2, 0)) → midpointtopleft
  15. Translate(midpoints, (deltax / 2, 0)) → midpoints'
  16. Translate(midpoints, (((-deltax)) / 2, 0)) → midpoints'_1
  17. Sequence(Polygon[Element[midpoints'_1, i], Element[midpoints', i], Element[midpointtopright, i], Element[midpointtopleft, i]], i, 1, n, 1) → list4
  18. Sum(list4) → h
  19. Translate(list1, ((-deltax), 0)) → list1'_1
  20. Translate(rightx, ((-deltax), 0)) → rightx'
  21. Sequence(Polygon[Element[rightx, i], Element[list1, i], Element[list1'_1, i], Element[rightx', i]], i, 1, n, 1) → rightrectanglesum
  22. Sum(rightrectanglesum) → j
  23. Intersect(k, p) → F
  24. Intersect(m, k) → G
  25. Polygon(D, E, G, F) → poly1, d_1, e, g_1, f_1
  26. Intersect(m, q) → H
  27. Intersect(p, q) → I
  28. Polygon(D, E, H, I) → poly2, d_2, e_1, h_1, i_1
  29. Intersect(p, r) → J
  30. Intersect(r, m) → K
  31. Polygon(D, E, K, J) → poly3, d_3, e_2, k_1, j_1
  32. Integral(f, a, b) → s