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AP 2007AB_1b 体积问题演示

AP 2007AB_1b 体积问题演示
几何画板及绘图内容加载中，请稍等……
2016-06-26 23:38
立体几何微积分积分参数方程几何作图
所属专题 Volumes in AP Calculus Volumes in AP Calculus
①本范例演示了如何通过旋转和积分计算旋转体的体积；②涉及的核心数学概念包括旋转体体积、积分计算、几何变换；③作图要点提炼自构造摘要，使用了 Sequence（数列）、Polygon（多边形）、Rotate（旋转）、Integral（积分）、RectangleSum（矩形和）、Surface（曲面）等 GeoGebra 命令/对象；④适合高中阶段学习微积分和立体几何的学生，用于理解旋转体体积的计算方法。

GeoGebra 命令及参数

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