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旋转体体积（壳法）

旋转体体积（壳法）
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2020-01-28 10:18
函数立体几何微积分积分
①本范例演示通过壳法计算旋转体体积的过程；②涉及核心数学概念包括积分、旋转体体积计算、函数图像；③构造要点：使用Line、Segment、CurveCartesian、Sequence、Textfield等命令绘制图形，涉及对象如e, aa, bb, n, ff, st, yDir, ss, f, Qss, Pss, PQ, gg, fInRotation, g, gInRotation, st2, lateralSurfAA, lateralSurfBB等；④适合高中及以上学生学习微积分和几何变换。

GeoGebra 命令及参数

  1. Line((0, 0, 0), (0, 1, 0)) → yDir
  2. Segment(Pss, Qss) → PQ
  3. CurveCartesian((t * cos(ss)), ff(t), (t * sin(ss)), t, aa, bb) → fInRotation
  4. CurveCartesian((t * cos(ss)), gg(t), (t * sin(ss)), t, aa, bb) → gInRotation
  5. Sequence(Segment[((aa * cos(k)), gg(aa), (aa * sin(k))), ((aa * cos(k)), ff(aa), (aa * sin(k)))], k, 0, (2 * π), st2) → lateralSurfAA
  6. Sequence(Segment[((bb * cos(k)), gg(bb), (bb * sin(k))), ((bb * cos(k)), ff(bb), (bb * sin(k)))], k, 0, (2 * π), st2) → lateralSurfBB
  7. Sequence(Segment[(t, gg(t), 0), (t, ff(t), 0)], t, aa, bb, st3) → region
  8. Sequence(Segment[((t * cos(ss)), gg(t), (t * sin(ss))), ((t * cos(ss)), ff(t), (t * sin(ss)))], t, aa, bb, st3) → regionInRotation
  9. Sequence(Circle[(0, gg(aa + (k * st)), 0), aa + (k * st), yDir], k, 0, n, st1) → lowerCircles
  10. Sequence(Circle[(0, ff(aa + (k * st)), 0), aa + (k * st), yDir], k, 0, n, st1) → upperCircles
  11. Textfield(ff) → InputBox1
  12. Textfield(gg) → InputBox2
  13. Textfield(aa) → InputBox3
  14. Textfield(bb) → InputBox4
  15. Textfield(st1) → InputBox5
  16. Textfield(st2) → InputBox6
  17. Sequence(Segment[(((tt + dx) * cos(k)), gg(tt), ((tt + dx) * sin(k))), (((tt + dx) * cos(k)), ff(tt), ((tt + dx) * sin(k)))], k, 0, (2 * π), stshell) → outerShell
  18. Sequence(Segment[((tt * cos(k)), gg(tt), (tt * sin(k))), ((tt * cos(k)), ff(tt), (tt * sin(k)))], k, 0, (2 * π), stshell) → innerShell
  19. Sequence(Circle[(0, ff(tt), 0), k, yDir], k, tt, tt + dx, stshell) → upperShellBase
  20. Textfield(stshell) → InputBox8
  21. Textfield(tt) → InputBox9
  22. Textfield(gg) → InputBox7