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已知交叉截面求体体积(正方形)

已知交叉截面求体体积(正方形)
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2016-06-27 00:17
多边形立体几何微积分积分轨迹
所属专题 Volumes in AP Calculus Volumes in AP Calculus
① 该范例通过GeoGebra演示已知正方形交叉截面时体的体积计算过程。② 核心数学概念包括微积分中的定积分积累法、已知截面求体体积的几何解释以及多边形/正方形的性质。③ 作图要点涉及Intersect、Polygon、Segment、Locus、Integral、If等GeoGebra命令，以及Square、A、B、C、D等几何对象的组合使用。④ 适合高中AP微积分或大学微积分入门阶段，帮助学生直观理解定积分在几何体积计算中的应用。

GeoGebra 命令及参数

  1. Intersect(f, g, (1.0328318883641743, 2.401108092689728)) → L
  2. Polygon(B, A, C, D) → Square, c, d, e, i
  3. Segment(A, B) → b
  4. Locus(D, s) → loc1
  5. Integral((ht(x))^(2), a, 2) → Integ
  6. If(togDF ≟ 1, "Show Instructions", "Show volume integral") → ToggleDirectionVolumeIntegral
  7. Locus(C, s) → loc2
  8. Angle(B, A, C) → α
  9. If(togS ≟ 0, "Show Surface", "Hide Surface") → ShowSurface
  10. Surface(u, f(u), (v * (f(u) - g(u))), u, xl, 2, v, 0, 1) → Back
  11. Surface(u, (v * g(u)) + ((1 - v) * f(u)), 0, u, xl, 2, v, 0, 1) → Base
  12. Surface(u, g(u), (v * (f(u) - g(u))), u, xl, 2, v, 0, 1) → Front
  13. Surface(u, (v * g(u)) + ((1 - v) * f(u)), f(u) - g(u), u, xl, 2, v, 0, 1) → Top
  14. Sequence(Polygon[(a + (k * h), g(a + ((k + p) * h)), 0), (a + (k * h), f(a + ((k + p) * h)), 0), (a + (k * h), f(a + ((k + p) * h)), ht(a + ((k + p) * h))), (a + (k * h), g(a + ((k + p) * h)), ht(a + ((k + p) * h)))], k, 0, m - 1) → SlicesLeft, {0.060143762897253684, 0.5232852472230439, 1.31086689094944, 2.126640129929901, 2.5868762004346118}
  15. Sequence(Area[Element[SlicesLeft, k]], k, 1, m) → Areas, {0.060143762897253684, 0.5232852472230439, 1.31086689094944, 2.126640129929901, 2.5868762004346118}
  16. Sequence(Surface[a + u, f(a + ((k + p) * h)), (v * ht(a + ((k + p) * h))), u, (k * h), ((k + 1) * h), v, 0, 1], k, 0, m - 1) → SlicesBack, {(1.0328318883641743 + u, f(1.0328318883641743 + ((0.5) * 0.12089601395447822)), (v * (ht(1.0328318883641743 + ((0.5) * 0.12089601395447822))))), (1.0328318883641743 + u, f(1.0328318883641743 + ((1 + 0.5) * 0.12089601395447822)), (v * (ht(1.0328318883641743 + ((1 + 0.5) * 0.12089601395447822))))), (1.0328318883641743 + u, f(1.0328318883641743 + ((2 + 0.5) * 0.12089601395447822)), (v * (ht(1.0328318883641743 + ((2 + 0.5) * 0.12089601395447822))))), (1.0328318883641743 + u, f(1.0328318883641743 + ((3 + 0.5) * 0.12089601395447822)), (v * (ht(1.0328318883641743 + ((3 + 0.5) * 0.12089601395447822))))), (1.0328318883641743 + u, f(1.0328318883641743 + ((4 + 0.5) * 0.12089601395447822)), (v * (ht(1.0328318883641743 + ((4 + 0.5) * 0.12089601395447822)))))}
  17. Sequence(Polygon[(a + (k * h), g(a + ((k + p) * h))), (a + ((k + 1) * h), g(a + ((k + p) * h))), (a + ((k + 1) * h), f(a + ((k + p) * h))), (a + (k * h), f(a + ((k + p) * h)))], k, 0, m - 1) → SlicesBase, {0.029648810900253064, 0.08745431199746978, 0.13841758867205822, 0.17630270854615393, 0.1944463541914181}
  18. Sequence(Surface[a + u, g(a + ((k + p) * h)), (v * ht(a + ((k + p) * h))), u, (k * h), ((k + 1) * h), v, 0, 1], k, 0, m - 1) → SlicesFront, {(1.0328318883641743 + u, g(1.0328318883641743 + ((0.5) * 0.12089601395447822)), (v * (ht(1.0328318883641743 + ((0.5) * 0.12089601395447822))))), (1.0328318883641743 + u, g(1.0328318883641743 + ((1 + 0.5) * 0.12089601395447822)), (v * (ht(1.0328318883641743 + ((1 + 0.5) * 0.12089601395447822))))), (1.0328318883641743 + u, g(1.0328318883641743 + ((2 + 0.5) * 0.12089601395447822)), (v * (ht(1.0328318883641743 + ((2 + 0.5) * 0.12089601395447822))))), (1.0328318883641743 + u, g(1.0328318883641743 + ((3 + 0.5) * 0.12089601395447822)), (v * (ht(1.0328318883641743 + ((3 + 0.5) * 0.12089601395447822))))), (1.0328318883641743 + u, g(1.0328318883641743 + ((4 + 0.5) * 0.12089601395447822)), (v * (ht(1.0328318883641743 + ((4 + 0.5) * 0.12089601395447822)))))}
  19. Sequence(Polygon[(a + ((k + 1) * h), g(a + ((k + p) * h)), 0), (a + ((k + 1) * h), f(a + ((k + p) * h)), 0), (a + ((k + 1) * h), f(a + ((k + p) * h)), ht(a + ((k + p) * h))), (a + ((k + 1) * h), g(a + ((k + p) * h)), ht(a + ((k + p) * h)))], k, 0, m - 1) → SlicesRight, {0.060143762897253684, 0.5232852472230439, 1.31086689094944, 2.126640129929901, 2.5868762004346118}
  20. Sequence(Surface[a + u, (v * g(a + ((k + p) * h))) + ((1 - v) * f(a + ((k + p) * h))), ht(a + ((k + p) * h)), u, (k * h), ((k + 1) * h), v, 0, 1], k, 0, m - 1) → SlicesTop, {(1.0328318883641743 + u, (v * (g(1.0328318883641743 + ((0.5) * 0.12089601395447822)))) + ((1 - v) * (f(1.0328318883641743 + ((0.5) * 0.12089601395447822)))), ht(1.0328318883641743 + ((0.5) * 0.12089601395447822))), (1.0328318883641743 + u, (v * (g(1.0328318883641743 + ((1 + 0.5) * 0.12089601395447822)))) + ((1 - v) * (f(1.0328318883641743 + ((1 + 0.5) * 0.12089601395447822)))), ht(1.0328318883641743 + ((1 + 0.5) * 0.12089601395447822))), (1.0328318883641743 + u, (v * (g(1.0328318883641743 + ((2 + 0.5) * 0.12089601395447822)))) + ((1 - v) * (f(1.0328318883641743 + ((2 + 0.5) * 0.12089601395447822)))), ht(1.0328318883641743 + ((2 + 0.5) * 0.12089601395447822))), (1.0328318883641743 + u, (v * (g(1.0328318883641743 + ((3 + 0.5) * 0.12089601395447822)))) + ((1 - v) * (f(1.0328318883641743 + ((3 + 0.5) * 0.12089601395447822)))), ht(1.0328318883641743 + ((3 + 0.5) * 0.12089601395447822))), (1.0328318883641743 + u, (v * (g(1.0328318883641743 + ((4 + 0.5) * 0.12089601395447822)))) + ((1 - v) * (f(1.0328318883641743 + ((4 + 0.5) * 0.12089601395447822)))), ht(1.0328318883641743 + ((4 + 0.5) * 0.12089601395447822)))}