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摆线下的面积

摆线下的面积
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2020-06-14 15:04
微积分积分参数方程几何作图
①本范例演示了如何计算摆线下的面积；②涉及的核心数学概念包括积分、微积分、极限和参数方程；③作图要点——从下方『构造摘要』提炼用到了哪些 GeoGebra 命令/对象，如 CurveCartesian, Point, Intersect, Circle, Segment, Midpoint, Sequence, Translate, Take；④适合的学段或用途——本范例适合大学微积分课程，用于演示摆线面积的计算方法和几何意义。

GeoGebra 命令及参数

  1. CurveCartesian((R * (i - sin(i))), (R * (1 - cos(i))), i, 0, Animation) → b
  2. Point(b) → L
  3. Intersect(xAxis, yAxis) → O
  4. Circle(C, R) → c
  5. Segment(C, A) → d
  6. Segment(D, B) → a
  7. Midpoint(a) → F
  8. Midpoint(O, B) → G
  9. Midpoint(B, E) → H
  10. Circle(D, If[Animation < 15, ((Animation - 10) * R / 5), R]) → q
  11. Midpoint(d) → Z
  12. Sequence(((R * (i - sin(i))), (R * (1 - cos(i)))), i, 0, (2 * π), (2 * π / 30)) → l1
  13. Sequence(Polygon[B, Element[l1, i], Element[l1, i + 1]], i, 1, Length[l1] / 3) → l2
  14. Sequence(Polygon[B, Element[l1, i], Element[l1, i + 1]], i, 1 + Length[l1] / 3, (2 * Length[l1] / 3)) → l3
  15. Circle(J, R) → e
  16. Sequence(Rotate[J + (R, 0), i, J], i, 0, (2 * π), (2 * π / 10)) → l5
  17. Sequence(Vector[Element[l5, i], Element[l1, i]], i, 1, 11) → l6
  18. Translate(J, Vector[(If[Animation < 10, 0, (Animation - 10) / 5] * Vector[J, B])]) → J'
  19. Sequence(Translate[Element[l5, i], Vector[(If[Animation < 10, 0, (Animation - 10) / 5] * Element[l6, i])]], i, 1, Length[l5]) → l7
  20. Sequence(Polygon[J', Element[l7, i], Element[l7, i + 1]], i, 1, Length[l7]) → l8
  21. Sequence(CircleSector[J, Element[l5, i], Element[l5, i + 1]], i, 1, Length[l5]) → l9
  22. Circle(I, R) → f
  23. Sequence(Rotate[I + (R, 0), i, I], i, 0, (2 * π), (2 * π / 10)) → l5b
  24. Take(l1, 11, 21) → l1b
  25. Sequence(Vector[Element[l5b, i], Element[l1b, i]], i, 1, Length[l1b]) → l6b
  26. Sequence(Translate[Element[l5b, i], Vector[(If[Animation < 10, 0, (Animation - 10) / 5] * Element[l6b, i])]], i, 1, Length[l5b]) → l7b
  27. Translate(I, Vector[(If[Animation < 10, 0, (Animation - 10) / 5] * Vector[I, B])]) → I'
  28. Sequence(Polygon[I', Element[l7b, i], Element[l7b, i + 1]], i, 1, Length[l7b]) → l8b
  29. Sequence(CircleSector[I, Element[l5b, i], Element[l5b, i + 1]], i, 1, Length[l5b]) → l9b
  30. Circle(M, R) → g
  31. Translate(M, Vector[(If[Animation < 10, 0, (Animation - 10) / 5] * Vector[M, B])]) → M'
  32. Take(l1, 21, 31) → l1c
  33. Sequence(Rotate[M + (R, 0), i, M], i, 0, (2 * π), (2 * π / 10)) → l5c
  34. Sequence(Vector[Element[l5c, i], Element[l1c, i]], i, 1, Length[l1c]) → l6c
  35. Sequence(Translate[Element[l5c, i], Vector[(If[Animation < 10, 0, (Animation - 10) / 5] * Element[l6c, i])]], i, 2, Length[l5c]) → l7c
  36. Sequence(CircleSector[M, Element[l5c, i], Element[l5c, i + 1]], i, 1, Length[l5c]) → l9c
  37. Sequence(Polygon[M', Element[l7c, i], Element[l7c, i + 1]], i, 2, Length[l7c]) → l8c
  38. Sequence(Polygon[B, Element[l1, i], Element[l1, i + 1]], i, (2 * Length[l1] / 3), Length[l1]) → l4