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测量体积3

测量体积3
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2016-05-30 17:19
函数立体几何微积分几何作图
①本范例演示如何通过构造正三角形和抛物线来测量三维图形的体积；②涉及的核心数学概念包括定积分、立体几何和函数图像；③作图要点：使用Point, Cube, Translate等GeoGebra命令创建点、立方体和变换；④适合高中阶段学习微积分和立体几何的学生使用。

GeoGebra 命令及参数

  1. Point(Circle[B, Distance[A, B], Segment[A, B]]) → C
  2. Point(Circle[J, Distance[I, J], Segment[I, J]]) → K
  3. Point(Circle[R, Distance[Q, R], Segment[Q, R]]) → S
  4. Point(Circle[H_1, Distance[G_1, H_1], Segment[G_1, H_1]]) → I_1
  5. Cube(G_1, H_1, I_1) → e_1, J_1, face11, facez, edge13, edge12, edge15, edge17, edge14, edge19, edge21, edge20, K_1, edge16, edge22, edge18, edge23, L_1, M_1, N_1, face7, face9, facey, facex
  6. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(0, 0), (2, 1)]) → f
  7. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(0, 0), (1, 0)]) → g
  8. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(0, 0), (1, 1, 1)]) → h
  9. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(0, 0), (1, 1)]) → e
  10. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(0, 0), (1, 1)]) → e'
  11. Translate(facex, Vector[(0, 0, 0), (0, -1, 0)]) → facex010
  12. Translate(facey, Vector[(0, 0), (1, 0, 0)]) → facey100
  13. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(2, 0)]) → a
  14. Translate(facez, Vector[(0, 0, 0), (0, -1, 0)]) → facez010
  15. Translate(facey, Vector[(0, 0, 1)]) → facey001
  16. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(1, 0, 1)]) → b
  17. Translate(facey, Vector[(0, -1, 0)]) → facey010
  18. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(1, -1)]) → c
  19. Translate(facex, Vector[(0, -3, 0)]) → facex030
  20. Translate(facex, Vector[(1, 0, 0)]) → facex100
  21. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(3, 1)]) → k
  22. Translate(facez, Vector[(0, -3, 0)]) → facez030
  23. Translate(facex, Vector[(0, 0, 1)]) → facex001
  24. Translate(facez, Vector[(1, 0, 0)]) → facez100
  25. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(2, 1, 1)]) → q
  26. Translate(facez, Vector[(0, 0, 1)]) → facez001
  27. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(1, 1, 2)]) → u
  28. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(0, 0), (2, 1)]) → f'
  29. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(2, 0)]) → a'
  30. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(0, 0), (1, 0)]) → g'
  31. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(3, 1)]) → k'
  32. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(1, -1)]) → c'
  33. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(0, 0), (1, 1, 1)]) → h'
  34. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(1, 0, 1)]) → b'
  35. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(2, 1, 1)]) → q'
  36. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[(1, 1, 2)]) → u'
  37. Translate(Cube[(0, 0), (1, 0), Point[Circle[(1, 0), Distance[(0, 0), (1, 0)], Segment[(0, 0), (1, 0)]]]], Vector[D]) → b_1
  38. Translate(Prism[Polygon[(-5, 2, 0), (-5, -1, 0), (-4, -1, 0), (-4, 0, 0), (-3, 0, 0), (-3, 1, 0), (-2, 1, 0), (-2, 2, 0)], (-5, 2, 1)], Vector[(0, 0)]) → d
  39. Translate(Prism[Polygon[(-5, 2, 1), (-5, 0, 1), (-4, 0, 1), (-4, 1, 1), (-3, 1, 1), (-3, 2, 1)], 1], Vector[(0, 0)]) → i
  40. Translate(Prism[Polygon[(-5, 2, 2), (-5, 1, 2), (-4, 1, 2), (-4, 2, 2)], 1], Vector[(0, 0)]) → j
  41. Translate(Prism[Polygon[(-5, 2, 0), (-5, -1, 0), (-4, -1, 0), (-4, 0, 0), (-3, 0, 0), (-3, 1, 0), (-2, 1, 0), (-2, 2, 0)], (-5, 2, 1)], Vector[(0, 0)]) → l
  42. Translate(Prism[Polygon[(-5, 2, 1), (-5, 0, 1), (-4, 0, 1), (-4, 1, 1), (-3, 1, 1), (-3, 2, 1)], 1], Vector[(0, 0)]) → m
  43. Translate(Prism[Polygon[(-5, 2, 2), (-5, 1, 2), (-4, 1, 2), (-4, 2, 2)], 1], Vector[(0, 0)]) → n
  44. Translate(Polygon[(-5, 0, 0), (-4, 0, 0), (-4, 0, 1), (-5, 0, 1)], Vector[(0, 0)]) → poly1
  45. Translate(Polygon[(-5, 1, 0), (-3, 1, 0), (-3, 1, 1), (-4, 1, 1), (-4, 1, 2), (-5, 1, 2)], Vector[(0, 0)]) → poly2
  46. Translate(Polygon[(-3, 1, 0), (-3, 2, 0), (-3, 2, 1), (-3, 1, 1)], Vector[(0, 0)]) → poly3
  47. Translate(Polygon[(-4, 0, 0), (-4, 2, 0), (-4, 2, 2), (-4, 1, 2), (-4, 1, 1), (-4, 0, 1)], Vector[(0, 0)]) → poly4