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利用圆柱壳法计算环面体积示例

利用圆柱壳法计算环面体积示例
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2016-03-15 11:07
函数立体几何微积分积分几何作图
本范例演示如何利用圆柱壳法计算环面体积。涉及的核心数学概念包括积分、体积计算和圆柱壳法。作图要点使用了GeoGebra的Sequence、Surface、Polygon、Segment等命令和对象，通过动态演示环面生成的过程，直观展示积分的几何意义。适合高中及以上学生学习和理解微积分中的体积计算方法。

GeoGebra 命令及参数

  1. Sequence(a + ((k - 0.5) * dx), k, 1, n) → midpts
  2. Sequence(a + (k * dx), k, 0, n - 1) → leftpts
  3. Sequence(a + (k * dx), k, 1, n) → rightpts
  4. Sequence(Polygon[(xk(k - 1), y_B(Element[evalpts, k])), (xk(k - 1), y_T(Element[evalpts, k])), (xk(k), y_T(Element[evalpts, k])), (xk(k), y_B(Element[evalpts, k]))], k, 1, n) → Rectangles
  5. Sequence((Element[evalpts, k], 0), k, 1, n) → evalptsplot
  6. Surface(xrot + ((xk(k - 1) - xrot) * cos(t)), h, ((xk(k - 1) - xrot) * sin(t)), t, 0, (2 * π), h, y_B(Element[evalpts, k]), y_T(Element[evalpts, k])) → ShellIn_k
  7. Surface(xrot + (r * cos(t)), y_T(Element[evalpts, k]), (r * sin(t)), t, 0, (2 * π), r, xk(k - 1) - xrot, xk(k) - xrot) → UpShell_k
  8. Surface(xrot + ((xk(k) - xrot) * cos(t)), h, ((xk(k) - xrot) * sin(t)), t, 0, (2 * π), h, y_B(Element[evalpts, k]), y_T(Element[evalpts, k])) → ShellOut_k
  9. Surface(xrot + (r * cos(t)), y_B(Element[evalpts, k]), (r * sin(t)), t, 0, (2 * π), r, xk(k - 1) - xrot, xk(k) - xrot) → ShellDown_k
  10. Sequence(Surface[xrot + ((xk(k - 1) - xrot) * cos(t)), h, ((xk(k - 1) - xrot) * sin(t)), t, 0, (2 * π), h, y_B(Element[evalpts, k]), y_T(Element[evalpts, k])], k, 1, n) → ShellsIn
  11. Sequence(Surface[xrot + (r * cos(t)), y_T(Element[evalpts, k]), (r * sin(t)), t, 0, (2 * π), r, xk(k - 1) - xrot, xk(k) - xrot], k, 1, n) → ShellsUp
  12. Sequence(Surface[xrot + (r * cos(t)), y_B(Element[evalpts, k]), (r * sin(t)), t, 0, (2 * π), r, xk(k - 1) - xrot, xk(k) - xrot], k, 1, n) → ShellsDown
  13. Sequence(Surface[xrot + ((xk(k) - xrot) * cos(t)), h, ((xk(k) - xrot) * sin(t)), t, 0, (2 * π), h, y_B(Element[evalpts, k]), y_T(Element[evalpts, k])], k, 1, n) → ShellsOut
  14. Surface(xrot + (cos((-t)) * (u - xrot)), y_B(u), (sin((-t)) * (u - xrot)), u, a, b, t, 0, RotationAngle) → Solid_B
  15. Surface(xrot + (cos((-t)) * (u - xrot)), y_T(u), (sin((-t)) * (u - xrot)), u, a, b, t, 0, RotationAngle) → Solid_T
  16. Surface(xrot + ((a - xrot) * cos((-t))), h, ((a - xrot) * sin((-t))), t, 0, RotationAngle, h, y_B(a), y_T(a)) → Solid_a
  17. Surface(xrot + ((b - xrot) * cos((-t))), h, ((b - xrot) * sin((-t))), t, 0, RotationAngle, h, y_B(b), y_T(b)) → Solid_b
  18. Surface(xrot + (cos((-RotationAngle)) * (u - xrot)), ((1 - w) * y_B(u)) + (w * y_T(u)), (sin((-RotationAngle)) * (u - xrot)), w, 0, 1, u, a, b) → Solid_e
  19. Surface(xrot + u - xrot, ((1 - w) * y_B(u)) + (w * y_T(u)), (0 * (u - xrot)), w, 0, 1, u, a, b) → region3d
  20. Polygon((xk(k - 1), y_B(Element[evalpts, k])), (xk(k - 1), y_T(Element[evalpts, k])), (xk(k), y_T(Element[evalpts, k])), (xk(k), y_B(Element[evalpts, k]))) → Rectangle_k, f, g, h, i
  21. Segment((0, 0), (xrot, 0)) → c