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正方形截面体积计算 by Mike May

正方形截面体积计算 by Mike May
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2016-03-19 11:36
立体几何微积分积分几何作图
①本范例演示通过正方形截面计算旋转体体积的过程；②涉及核心数学概念包括体积积分、立体几何、二次函数；③构造要点使用了Intersect（相交）、Segment（线段）、Locus（轨迹）、Polygon（多边形）等GeoGebra命令；④适合高中及以上学生用于理解体积积分与立体几何的综合应用。

GeoGebra 命令及参数

  1. Intersect(log(x), 5 - x, 3, 4) → E
  2. Segment(A, B) → e
  3. Locus(C, s) → loc1
  4. If(togDF ≟ 1, "Show Instructions", "Show volume integral") → ToggleDirectionVolumeIntegral
  5. If(togS ≟ 1, "Hide surface", "Show surface") → ShowSurface
  6. Locus(D, s) → loc2
  7. Polygon(A, B, C, D) → Square, a, b, c, d
  8. If(togB ≟ 1, "HIde boundary square", "Show boundary square") → BoundarySquare
  9. Surface(u, (v * f(u)) + ((1 - v) * g(u)), 0, u, 1, 5, v, 0, 1) → Base
  10. Surface(u, (v * f(u)) + ((1 - v) * g(u)), g(u), u, 1, 5, v, 0, 1) → Top
  11. Surface(u, 0, (v * g(u)), u, 1, aa, v, 0, 1) → SurfaceFront_1
  12. Surface(u, 0, (v * g(u)), u, aa, 5, v, 0, 1) → SurfaceFront_2
  13. Surface(u, g(u), (v * (g(u) - f(u))), v, 0, 1, u, 1, aa) → SurfaceBack_1
  14. Surface(u, g(u), (v * (g(u) - f(u))), v, 0, 1, u, aa, 5) → SurfaceBack_2
  15. Sequence(Polygon[(1 + ((k + 1) * h), 0, 0), (1 + ((k + 1) * h), g(1 + ((k + p) * h)), 0), (1 + ((k + 1) * h), g(1 + ((k + p) * h)), g(1 + ((k + p) * h))), (1 + ((k + 1) * h), 0, g(1 + ((k + p) * h)))], k, 0, m - 1) → SlicesRight
  16. Sequence(Polygon[(1 + (k * h), 0, 0), (1 + (k * h), g(1 + ((k + p) * h)), 0), (1 + (k * h), g(1 + ((k + p) * h)), g(1 + ((k + p) * h))), (1 + (k * h), 0, g(1 + ((k + p) * h)))], k, 0, m - 1) → SlicesLeft
  17. Sequence(Surface[u, 0, (v * g(1 + ((k + p) * h))), u, 1 + (k * h), 1 + ((k + 1) * h), v, 0, 1], k, 0, m - 1) → SlicesFront
  18. Sequence(Surface[u, g(1 + ((k + p) * h)), (v * g(1 + ((k + p) * h))), u, 1 + (k * h), 1 + ((k + 1) * h), v, 0, 1], k, 0, m - 1) → SlicesBack
  19. Sequence(Surface[u, (v * g(1 + ((k + p) * h))), g(1 + ((k + p) * h)), u, 1 + (k * h), 1 + ((k + 1) * h), v, 0, 1], k, 0, m - 1) → SlicesTop
  20. Sequence(Polygon[(1 + (k * h), 0), (1 + (k * h), g(1 + ((k + p) * h))), (1 + ((k + 1) * h), g(1 + ((k + p) * h))), (1 + ((k + 1) * h), 0)], k, 0, m - 1) → SlicesBase
  21. Sequence(Area[Element[SlicesLeft, k]], k, 1, m) → Areas
  22. Polygon((bs, 0, 0), (bs, g(bs), 0), (bs, g(bs), g(bs)), (bs, 0, g(bs))) → BoundSquare, i, j, k, l