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旋转体体积计算示例

旋转体体积计算示例
几何画板及绘图内容加载中，请稍等……
2015-07-24 08:21
函数立体几何微积分几何作图
该范例演示了如何使用圆盘法计算函数y=f(x)绕水平轴旋转形成的旋转体体积。核心数学概念包括微积分中的积分思想、几何中的体积公式以及函数图像的变换。作图要点涉及使用GeoGebra命令如Point、Intersect、Surface、Polygon等来构造旋转体模型。适合高中生或大学低年级学生用于学习微积分和立体几何。

GeoGebra 命令及参数

  1. Point(xAxis) → A
  2. Point(xAxis) → B
  3. Intersect(f, b, (5, 3.23606797749979)) → D
  4. Intersect(f, a, (1, 2)) → C
  5. Segment(A, C) → d
  6. Segment(B, D) → e
  7. Sequence((x(A) + (k * Δx), (f(x(A) + (k * Δx)) * cos(ϕ)), (f(x(A) + (k * Δx)) * sin(ϕ))), k, 0, n - 1) → tracepoints
  8. Textfield(f) → inputBox1
  9. Textfield(axis) → inputBox2
  10. Sequence(Surface[x(A) + (u * Δx), axis + ((f(x(A) + (k * Δx)) - axis) * cos(v)), ((f(x(A) + (k * Δx)) - axis) * sin(v)), u, k, k + 1, v, 0, ϕ], k, 0, n - 1) → surfaces
  11. Surface(x(A) + (u * Δx), axis + ((f(x(A) + ((i - 1) * Δx)) - axis) * cos(v)), ((f(x(A) + ((i - 1) * Δx)) - axis) * sin(v)), u, i - 1, i, v, 0, ϕ) → surface_{i}
  12. Sequence(Polygon[(x(A) + (k * Δx), axis), (x(A) + (k * Δx), f(x(A) + (k * Δx))), (x(A) + ((k + 1) * Δx), f(x(A) + (k * Δx))), (x(A) + ((k + 1) * Δx), axis)], k, 0, n - 1) → rectangles
  13. Polygon((x(A) + ((i - 1) * Δx), axis, 0), (x(A) + ((i - 1) * Δx), axis + ((f(x(A) + ((i - 1) * Δx)) - axis) * cos(ϕ)), ((f(x(A) + ((i - 1) * Δx)) - axis) * sin(ϕ))), (x(A) + (i * Δx), axis + ((f(x(A) + ((i - 1) * Δx)) - axis) * cos(ϕ)), ((f(x(A) + ((i - 1) * Δx)) - axis) * sin(ϕ))), (x(A) + (i * Δx), axis, 0)) → face_{i}, c, j, k, l
  14. Sequence(CircumcircleSector[(x(A) + (k * Δx), f(x(A) + ((i - 1) * Δx)), 0), (x(A) + (k * Δx), axis + ((f(x(A) + ((i - 1) * Δx)) - axis) * cos(ϕ / 2)), ((f(x(A) + ((i - 1) * Δx)) - axis) * sin(ϕ / 2))), (x(A) + (k * Δx), axis + ((f(x(A) + ((i - 1) * Δx)) - axis) * cos(ϕ)), ((f(x(A) + ((i - 1) * Δx)) - axis) * sin(ϕ)))], k, i - 1, i) → endcaps_{i}
  15. Sequence(Polygon[(x(A) + ((k - 1) * Δx), axis, 0), (x(A) + ((k - 1) * Δx), axis + ((f(x(A) + ((k - 1) * Δx)) - axis) * cos(ϕ)), ((f(x(A) + ((k - 1) * Δx)) - axis) * sin(ϕ))), (x(A) + (k * Δx), axis + ((f(x(A) + ((k - 1) * Δx)) - axis) * cos(ϕ)), ((f(x(A) + ((k - 1) * Δx)) - axis) * sin(ϕ))), (x(A) + (k * Δx), axis, 0)], k, 1, n) → faces
  16. Sequence(CircumcircleSector[(x(A) + (k * Δx), f(x(A) + (k * Δx)), 0), (x(A) + (k * Δx), axis + ((f(x(A) + (k * Δx)) - axis) * cos(ϕ / 2)), ((f(x(A) + (k * Δx)) - axis) * sin(ϕ / 2))), (x(A) + (k * Δx), axis + ((f(x(A) + (k * Δx)) - axis) * cos(ϕ)), ((f(x(A) + (k * Δx)) - axis) * sin(ϕ)))], k, 0, n - 1) → endcaps_{left}
  17. Sequence(CircumcircleSector[(x(A) + (k * Δx), f(x(A) + ((k - 1) * Δx)), 0), (x(A) + (k * Δx), axis + ((f(x(A) + ((k - 1) * Δx)) - axis) * cos(ϕ / 2)), ((f(x(A) + ((k - 1) * Δx)) - axis) * sin(ϕ / 2))), (x(A) + (k * Δx), axis + ((f(x(A) + ((k - 1) * Δx)) - axis) * cos(ϕ)), ((f(x(A) + ((k - 1) * Δx)) - axis) * sin(ϕ)))], k, 1, n) → endcaps_{right}
  18. Polygon((x(A) + ((i - 1) * Δx), axis), (x(A) + ((i - 1) * Δx), f(x(A) + ((i - 1) * Δx))), (x(A) + (i * Δx), f(x(A) + ((i - 1) * Δx))), (x(A) + (i * Δx), axis)) → rectangle_{i}, g, m, p, q
  19. Sequence(Cylinder[(x(A) + (k * Δx), axis, 0), (x(A) + ((k + 1) * Δx), axis, 0), f(x(A) + (k * Δx)) - axis], k, 0, n - 1) → cylinders
  20. Surface(t, axis + ((f(t) - axis) * cos(u)), ((f(t) - axis) * sin(u)), t, x(A), x(B), u, 0, ϕ) → surface_{whole}
  21. CircumcircleSector((x(A), f(x(A)), 0), (x(A), axis + ((f(x(A)) - axis) * cos(ϕ / 2)), ((f(x(A)) - axis) * sin(ϕ / 2))), (x(A), axis + ((f(x(A)) - axis) * cos(ϕ)), ((f(x(A)) - axis) * sin(ϕ)))) → endcap_{A}
  22. CircumcircleSector((x(B), f(x(B)), 0), (x(B), axis + ((f(x(B)) - axis) * cos(ϕ / 2)), ((f(x(B)) - axis) * sin(ϕ / 2))), (x(B), axis + ((f(x(B)) - axis) * cos(ϕ)), ((f(x(B)) - axis) * sin(ϕ)))) → endcap_{B}

AI 绘图提示语参考

画一个函数图像，并在其上标出两个点，通过这两个点和函数图像的交点构造两个线段。利用这些线段和函数图像，绘制一系列矩形和圆弧扇形，模拟旋转体的横截面和侧面。最后将这些图形组合起来，展示一个旋转体的体积计算过程，包含底面、侧面和顶面。
以上提示语为根据原始绘图命令整理而来，仅供参考。