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旋转体体积可视化 by Lila Roberts

旋转体体积可视化 by Lila Roberts
几何画板及绘图内容加载中，请稍等……
2016-01-09 01:58
函数解析几何立体几何微积分积分
所属专题 Visualizing Solids of Revolution and Volumes
①本范例演示由函数f(x)=x²、x轴及x=4所围区域旋转生成的旋转体；②涉及核心数学概念：旋转体体积、积分、解析几何；③作图要点：使用了CurveCartesian、Polygon、Surface等GeoGebra命令构造函数曲线、多边形和曲面，通过curverev、washerpolyrev等命令实现旋转体和环形薄片的可视化；④适合高中及以上学生及微积分学习者用于理解体积计算和三维图形表示。

GeoGebra 命令及参数

  1. CurveCartesian(t, 0, t^(2), t, 0, 2) → fncurve
  2. Polygon((1, 0, 0), (1.2, 0, 0), (1.2, 0, 1), (1, 0, 1)) → shellpoly, h, i, j, k
  3. Surface((t * cos(s)), (t * sin(s)), t^(2), t, 0, 2, s, 0, theta) → curverev
  4. Polygon((1, 0, 1), (2, 0, 1), (2, 0, 1.2), (1, 0, 1.2)) → washerpoly, r, s, t, a_1
  5. Polygon((cos(theta), sin(theta), 0), ((1.2 * cos(theta)), (1.2 * sin(theta)), 0), ((1.2 * cos(theta)), (1.2 * sin(theta)), 1), (cos(theta), sin(theta), 1)) → shellpolyrev, l, m, n, p
  6. Polygon((cos(theta), sin(theta), 1), ((2 * cos(theta)), (2 * sin(theta)), 1), ((2 * cos(theta)), (2 * sin(theta)), 1.2), (cos(theta), sin(theta), 1.2)) → washerpolyrev, b_1, c_1, d_1, e_1
  7. CurveCartesian((t * cos(theta)), (t * sin(theta)), t^(2), t, 0, 2) → fncurverev
  8. CurveCartesian(2, 0, t, t, 0, 4) → vertcurve
  9. CurveCartesian((2 * cos(theta)), (2 * sin(theta)), t, t, 0, 4) → vertcurverev
  10. CurveCartesian(t, 0, 0, t, 0, 2) → horizaxis
  11. CurveCartesian((t * cos(theta)), (t * sin(theta)), 0, t, 0, 2) → horizaxisrev
  12. Surface((t * cos(s)), (t * sin(s)), 0, t, 0, 2, s, 0, theta) → horizaxisrev_1
  13. Surface((2 * cos(s)), (2 * sin(s)), t, t, 0, 4, s, 0, theta) → verticalrev
  14. Surface(cos(s), sin(s), t^(2), t, 0, 1, s, 0, theta) → f
  15. Surface((1.2 * cos(s)), (1.2 * sin(s)), t^(2), t, 0, 1, s, 0, theta) → g
  16. Surface((r * cos(s)), (r * sin(s)), 0, r, 0.5, 0.7, s, 0, theta) → o
  17. Surface(cos(s), sin(s), t, t, 1, 1.2, s, 0, theta) → w
  18. Surface((t * cos(s)), (t * sin(s)), 1, t, 1, 1.2, s, 0, theta) → v
  19. Surface((t * cos(s)), (t * sin(s)), 1.2, t, 1, 2, s, 0, theta) → u
  20. Surface((t * cos(s)), (t * sin(s)), 0, t, 1, 1.2, s, 0, theta) → shellaxisrev
  21. Surface((r * cos(s)), (r * sin(s)), 1, r, 1, 1.2, s, 0, theta) → q
  22. Surface((2 * cos(s)), (2 * sin(s)), t, t, 1, 1.2, s, 0, theta) → b