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旋转体可视化与体积计算

旋转体可视化与体积计算
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2016-01-09 03:26
函数立体几何
所属专题 Visualizing Solids of Revolution and Volumes
该范例演示了由函数f(x)=√x和g(x)=x围成的区域绕x轴旋转生成的旋转体。核心数学概念包括旋转体、体积计算。使用了Polygon、CurveCartesian、Surface等GeoGebra命令。适合高中数学教学，帮助学生理解旋转体的生成过程和体积计算方法。

GeoGebra 命令及参数

  1. Polygon((0.5^(2), 0, 0.5), (0.5^(2), 0, 0.6), (sqrt(0.5), 0, 0.6), (sqrt(0.5), 0, 0.5)) → washerpoly, e, f, g, h
  2. Polygon(((0.5 * cos(theta)), (0.5 * sin(theta)), 0.5^(2)), ((0.6 * cos(theta)), (0.6 * sin(theta)), 0.5^(2)), ((0.6 * cos(theta)), (0.6 * sin(theta)), sqrt(0.5)), ((0.5 * cos(theta)), (0.5 * sin(theta)), sqrt(0.5))) → shellpolyrev, a, b, c, d
  3. Polygon(((0.5^(2) * cos(theta)), (0.5^(2) * sin(theta)), 0.5), ((0.5^(2) * cos(theta)), (0.5^(2) * sin(theta)), 0.6), ((sqrt(0.5) * cos(theta)), (sqrt(0.5) * sin(theta)), 0.6), ((sqrt(0.5) * cos(theta)), (sqrt(0.5) * sin(theta)), 0.5)) → washerpolyrev, i, j, k, l
  4. CurveCartesian(t, 0, sqrt(t), t, 0, 1) → topcurve
  5. CurveCartesian((t * cos(theta)), (t * sin(theta)), t^(2), t, 0, 1) → fncurverev
  6. CurveCartesian(t, 0, t^(2), t, 0, 1) → fncurve
  7. CurveCartesian((t * cos(theta)), (t * sin(theta)), sqrt(t), t, 0, 1) → topcurverev_1
  8. Surface((0.5 * cos(s)), (0.5 * sin(s)), t, t, 0.5^(2), sqrt(0.5), s, 0, theta) → shellinsiderev
  9. Surface((0.6 * cos(s)), (0.6 * sin(s)), t, t, 0.5^(2), sqrt(0.5), s, 0, theta) → shelloutsiderev
  10. Surface((t * cos(s)), (t * sin(s)), 0.5^(2), t, 0.5, 0.6, s, 0, theta) → shellbottomrev
  11. Surface((t * cos(s)), (t * sin(s)), t^(2), t, 0, 1, s, 0, theta) → bottomcurverev
  12. Surface((t * cos(s)), (t * sin(s)), sqrt(t), t, 0, 1, s, 0, theta) → topcurverev
  13. Surface((t * cos(s)), (t * sin(s)), 0.5, t, 0.5^(2), sqrt(0.5), s, 0, theta) → washerbottom
  14. Surface((t * cos(s)), (t * sin(s)), 0.6, t, 0.5^(2), sqrt(0.5), s, 0, theta) → washertop
  15. Surface((sqrt(0.5) * cos(s)), (sqrt(0.5) * sin(s)), t, t, 0.5, 0.6, s, 0, theta) → washeroutside
  16. Surface((0.5^(2) * cos(s)), (0.5^(2) * sin(s)), t, t, 0.5, 0.6, s, 0, theta) → washerinside
  17. Surface((t * cos(s)), (t * sin(s)), sqrt(0.5), t, 0.5, 0.6, s, 0, theta) → shelltoprev
  18. Polygon((0.5, 0, 0.5^(2)), (0.6, 0, 0.5^(2)), (0.6, 0, sqrt(0.5)), (0.5, 0, sqrt(0.5))) → shellpoly, p, q, r, s