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旋转抛物面体积计算

旋转抛物面体积计算
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2015-07-16 23:15
立体几何微积分积分
①本范例演示了两个二次函数绕y=-3旋转形成的旋转体体积计算；②涉及核心数学概念包括旋转体体积、圆环面积法（垫圈法）、微积分积分应用；③构造过程中运用了GeoGebra命令Sequence（数列）、Rotate（旋转）、RectangleSum（矩形求和）、Surface（曲面）、Polygon（多边形）、Integral（积分）；④适合高中及以上学生学习微积分体积计算方法及几何可视化。

GeoGebra 命令及参数

  1. Sequence(Polygon[(-1 + (k * h), g(-1 + ((k + p) * h))), (-1 + ((k + 1) * h), g(-1 + ((k + p) * h))), (-1 + ((k + 1) * h), f(-1 + ((k + p) * h))), (-1 + (k * h), f(-1 + ((k + p) * h)))], k, 0, m - 1) → Rectangles
  2. Rotate(Rectangles, (-α), a) → Rectangles'
  3. RectangleSum(A, -1, -1 + (m * h), m, p) → VolWashers
  4. Surface(-1 + (DK * h), -3 + ((u + 3) * cos(β)), ((u + 3) * sin(β)), u, g(-1 + ((DK + p) * h)), f(-1 + ((DK + p) * h)), β, 0, α) → SideLeft1
  5. Surface(-1 + ((DK + 1) * h), -3 + ((u + 3) * cos(β)), ((u + 3) * sin(β)), u, g(-1 + ((DK + p) * h)), f(-1 + ((DK + p) * h)), β, 0, α) → SideRight1
  6. Surface(u, -3 + ((f(-1 + ((DK + p) * h)) + 3) * cos(β)), ((f(-1 + ((DK + p) * h)) + 3) * sin(β)), u, -1 + (DK * h), -1 + ((DK + 1) * h), β, 0, α) → Outer1
  7. Surface(u, -3 + ((g(-1 + ((DK + p) * h)) + 3) * cos(β)), ((g(-1 + ((DK + p) * h)) + 3) * sin(β)), u, -1 + (DK * h), -1 + ((DK + 1) * h), β, 0, α) → Inner1
  8. Polygon((-1 + (DK * h), g(-1 + ((DK + p) * h))), (-1 + ((DK + 1) * h), g(-1 + ((DK + p) * h))), (-1 + ((DK + 1) * h), f(-1 + ((DK + p) * h))), (-1 + (DK * h), f(-1 + ((DK + p) * h)))) → Rectangle1, b, c, d, e
  9. Rotate(Rectangle1, (-α), a) → Rectangle1'
  10. Integral((π * ((f(x) + 3)^(2) - (g(x) + 3)^(2))), -1, 2) → VolInteg
  11. Rotate(f_1, (-α), a) → i
  12. Rotate(g_1, (-α), a) → j
  13. Surface(u, -3 + ((f(u) + 3) * cos(β)), ((f(u) + 3) * sin(β)), u, -1, 2, β, 0, (2 * π)) → Outer
  14. Surface(u, -3 + ((g(u) + 3) * cos(β)), ((g(u) + 3) * sin(β)), u, -1, 2, β, 0, (2 * π)) → Inner
  15. Sequence(Surface[-1 + ((k + u) * h), -3 + ((3 + f(-1 + ((k + p) * h))) * cos(β)), ((3 + f(-1 + ((k + p) * h))) * sin(β)), u, 0, 1, β, 0, α], k, 0, m - 1) → Outers
  16. Sequence(Surface[-1 + ((k + u) * h), -3 + ((3 + g(-1 + ((k + p) * h))) * cos(β)), ((3 + g(-1 + ((k + p) * h))) * sin(β)), u, 0, 1, β, 0, α], k, 0, m - 1) → Inners
  17. Sequence(Surface[-1 + (k * h), -3 + ((3 + (v * g(-1 + ((k + p) * h))) + ((1 - v) * f(-1 + ((k + p) * h)))) * cos(β)), ((3 + (v * g(-1 + ((k + p) * h))) + ((1 - v) * f(-1 + ((k + p) * h)))) * sin(β)), v, 0, 1, β, 0, α], k, 0, m - 1) → SidesLeft
  18. Sequence(Surface[-1 + ((k + 1) * h), -3 + ((3 + (v * g(-1 + ((k + p) * h))) + ((1 - v) * f(-1 + ((k + p) * h)))) * cos(β)), ((3 + (v * g(-1 + ((k + p) * h))) + ((1 - v) * f(-1 + ((k + p) * h)))) * sin(β)), v, 0, 1, β, 0, α], k, 0, m - 1) → SidesRight