积极 AI Logo 积极 AI

圆盘法求旋转体体积 by Mike May (2)

圆盘法求旋转体体积 by Mike May (2)
几何画板及绘图内容加载中，请稍等……
2016-03-20 02:59
立体几何微积分积分
所属专题 AP Calculus volume problems
本例演示2010年AP考试第4b题，通过圆盘法计算旋转体体积。涉及的核心数学概念包括旋转体体积、抛物线。GeoGebra中使用了Sequence, Polygon, Rotate, Integral, Surface, Segment, RectangleSum等命令。适合高中数学教学和学习。

GeoGebra 命令及参数

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

AI 绘图提示语参考

绘制一个旋转体的体积计算图，展示使用圆环法求解。图中包含若干个矩形和抛物线旋转形成的曲面，通过积分计算体积。首先绘制基础的矩形序列，然后旋转并组合这些矩形形成圆环，接着绘制内外曲面的旋转体，最后通过积分线段和曲面组合展示整个旋转体，体现体积计算过程。
以上提示语为根据原始绘图命令整理而来，仅供参考。