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利用壳层法计算体积

利用壳层法计算体积
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2016-03-18 09:10
立体几何微积分积分
这个范例演示了如何使用壳层法计算由曲线绕轴旋转形成的体积。涉及的核心数学概念包括微积分中的积分计算、旋转体体积公式以及壳层法应用。作图要点包括使用Segment、Polygon、Rotate、Integral、Sequence、Surface、RectangleSum等GeoGebra命令和对象来构建旋转体模型、生成壳层并计算积分。适合高中及以上学习微积分的学生，特别是需要理解旋转体体积计算方法的学生。

GeoGebra 命令及参数

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