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定积分作为黎曼和的极限

定积分作为黎曼和的极限
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2017-04-11 23:54
函数微积分积分极限几何作图
这个范例演示了如何通过黎曼和的极限来计算定积分，直观展示了积分与面积的关系。涉及的核心数学概念包括定积分、黎曼和、梯形法、向量等。作图要点使用了GeoGebra的If、Sequence、Integral、LaTeX、Segment、FractionText、Midpoint、Vector、Distance、Intersect、Point等命令和对象，通过滑块动画展示积分过程。适合高中及以上学生学习和理解微积分基本概念。

GeoGebra 命令及参数

  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