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旋转体壳层法演示

旋转体壳层法演示
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2015-09-13 13:21
函数三角函数立体几何微积分积分
本范例演示通过壳层法计算旋转体体积的过程。涉及的核心数学概念包括定积分、旋转、立体几何体积、二次函数、自然数、三角函数等。作图要点使用了Point、CurveCartesian、Integral、Segment、Line、Surface、Sequence等GeoGebra命令与对象，如yMax, yMin, xMin, xMax, UpperFunction, ShellBar等，通过滑块和动态演示展示体积积分计算。适合高中及以上学生及微积分教学使用。

GeoGebra 命令及参数

  1. Point(y = 0) → R
  2. Point(xAxis) → RightEnd
  3. Point(xAxis) → LeftEnd
  4. CurveCartesian(t, (0.8 * yMin) + (0.2 * yMax), t, x(LeftEnd), x(RightEnd)) → ShellBar
  5. Point(ShellBar) → ShellPoint
  6. CurveCartesian(t, UpperFunction(t), t, x(LeftEnd), x(RightEnd)) → UpperCurve
  7. Textfield(UpperFunction) → UpperFunctionBox
  8. Integral(ShellArea, x(LeftEnd), x(ShellPoint)) → VolumeToDisk
  9. Integral(ShellArea, x(LeftEnd), x(RightEnd)) → FullVolume
  10. Textfield(xMin) → xMinBox
  11. Textfield(xMax) → xMaxBox
  12. Textfield(yMin) → yMinBox
  13. Textfield(yMax) → yMaxtBox
  14. CurveCartesian(t, LowerFunction(t), t, x(LeftEnd), x(RightEnd)) → LowerCurve
  15. Segment((x(ShellPoint), LowerFunction(x(ShellPoint))), (x(ShellPoint), UpperFunction(x(ShellPoint)))) → LineToCurve
  16. Textfield(LowerFunction) → LowerFunctionBox
  17. Line(R, R + (0, 1)) → AxisOfRotation
  18. Surface(x(R) + (cos(t) * (x(ShellPoint) - x(R))), (s * UpperFunction(x(ShellPoint))) + ((1 - s) * LowerFunction(x(ShellPoint))), (sin(t) * (x(ShellPoint) - x(R))), s, 0, 1, t, 0, 6.28319) → IntegrationDisk
  19. Surface(x(R) + (cos(t) * (s - x(R))), UpperFunction(s), (sin(t) * (s - x(R))), s, x(LeftEnd), x(RightEnd), t, 6.2832 - RotationAngleShown, 6.2832) → SurfaceRev
  20. Surface(x(R) + (cos(t) * (s - x(R))), LowerFunction(s), (sin(t) * (s - x(R))), s, x(LeftEnd), x(RightEnd), t, 6.2832 - RotationAngleShown, 6.2832) → SurfaceRevInner
  21. Sequence(CurveCartesian[x(R) + (cos((((n + 0.5) * 2) * π / 12)) * (t - x(R))), UpperFunction(t), (sin((((n + 0.5) * 2) * π / 12)) * (t - x(R))), t, x(LeftEnd), x(RightEnd)], n, 0, 11) → WireFrameRibs
  22. Sequence(CurveCartesian[x(R) + (cos((((n + 0.5) * 2) * π / 12)) * (t - x(R))), LowerFunction(t), (sin((((n + 0.5) * 2) * π / 12)) * (t - x(R))), t, x(LeftEnd), x(RightEnd)], n, 0, 11) → WireFrameRibsInner