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圆盘法求旋转体体积 by Mike May

圆盘法求旋转体体积 by Mike May
几何画板及绘图内容加载中，请稍等……
2016-03-20 09:53
微积分积分几何作图
本示例演示2006年AP考试第1b题，通过圆盘法计算旋转体体积。涉及的核心数学概念包括积分和旋转体体积。使用了Intersect, Polygon, Rotate, Integral, Sequence, Surface, RectangleSum等GeoGebra命令。适合高中微积分课程使用。

GeoGebra 命令及参数

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

AI 绘图提示语参考

绘制一个围绕x轴旋转的立体图形，该图形由两个函数f(x)和g(x)在区间[a, b]上生成的外环和内环构成。首先绘制两个函数的图像，然后围绕x轴旋转生成外环和内环的表面，接着绘制旋转后的外环和内环的侧面，最后计算并显示该立体图形的体积。
以上提示语为根据原始绘图命令整理而来，仅供参考。