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2011年AP微积分AB卷B第3c题：体积问题

2011年AP微积分AB卷B第3c题：体积问题
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2016-06-26 22:24
微积分积分参数方程几何作图滑块动画
所属专题 Volumes in AP Calculus Volumes in AP Calculus
本示例演示了2011年AP微积分AB卷B第3c题中的体积问题。核心数学概念包括微积分中的积分。通过使用GeoGebra命令如Sequence, Intersect, Segment, Locus, Integral等，构建了涉及相似三角形和参数曲线的几何图形。适合高中微积分课程教学和学习。

GeoGebra 命令及参数

  1. Sequence(Polygon[(fI(((k + p) * h)), (k * h), 0), (g(((k + p) * h)), (k * h), 0), (g(((k + p) * h)), (k * h), ((2 * (k + p)) * h)), (fI(((k + p) * h)), (k * h), ((2 * (k + p)) * h))], k, 0, m - 1) → SlicesFront
  2. Intersect(f, g, 3, 5) → E
  3. Segment(A, B) → e
  4. Locus(C, s) → loc1
  5. Integral(((2 * y) * (6 - y - y^(2))), 0, 2) → Integ
  6. If(togDF ≟ 1, "Show Instructions", "Show volume integral") → ToggleDirectionVolumeIntegral
  7. If(togS ≟ 1, "Hide surface", "Show surface") → ShowSurface
  8. Locus(D, s) → loc2
  9. Polygon(A, B, C, D) → Rect, a, b, c, d
  10. Angle(B, A, D) → α
  11. Surface(u^(2), u, (v * (2 * u)), u, 0, 2, v, 0, 1) → SurfaceLeft
  12. Surface(6 - u, u, (v * (2 * u)), u, 0, 2, v, 0, 1) → SurfaceRight
  13. Surface((v * u^(2)) + ((1 - v) * (6 - u)), u, 0, u, 0, 2, v, 0, 1) → Base
  14. Surface((v * u^(2)) + ((1 - v) * (6 - u)), u, (2 * u), u, 0, 2, v, 0, 1) → SurfaceFront
  15. Sequence(Surface[fI(((k + p) * h)), (k * h) + (v * h), ((u * (k + p)) * h), v, 0, 1, u, 0, 2], k, 0, m - 1) → SlicesLeft
  16. Sequence(Surface[u, v, ((2 * (k + p)) * h), v, (k * h), ((k + 1) * h), u, fI(((k + p) * h)), g(((k + p) * h))], k, 0, m - 1) → SlicesTop
  17. Sequence(Surface[g(((k + p) * h)), (k * h) + (v * h), ((u * (k + p)) * h), v, 0, 1, u, 0, 2], k, 0, m - 1) → SlicesRight
  18. Sequence(Area[Element[SlicesFront, k]], k, 1, m) → Areas
  19. Sequence(Polygon[(fI(((k + p) * h)), (k * h), 0), (g(((k + p) * h)), (k * h), 0), (g(((k + p) * h)), ((k + 1) * h), 0), (fI(((k + p) * h)), ((k + 1) * h), 0)], k, 0, m - 1) → SlicesBase
  20. Sequence(Polygon[(fI(((k + p) * h)), ((k + 1) * h), 0), (g(((k + p) * h)), ((k + 1) * h), 0), (g(((k + p) * h)), ((k + 1) * h), ((2 * (k + p)) * h)), (fI(((k + p) * h)), ((k + 1) * h), ((2 * (k + p)) * h))], k, 0, m - 1) → SlicesBack