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旋转体体积的积分估算与可视化

旋转体体积的积分估算与可视化
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2018-05-06 09:11
函数立体几何微积分积分滑块动画
① 本范例通过GeoGebra动态演示了利用定积分计算函数曲线绕x轴旋转所得旋转体体积的过程，并同时展示了左矩形、右矩形估算方法。② 核心数学概念包括定积分、旋转体体积、黎曼和、圆柱薄壳法以及函数图像与旋转曲面的三维可视化。③ 作图要点：使用了Textfield、Integral、Sequence、Sum、Surface、CircumcircleSector等命令，构建了上下限输入框、定积分面积、左右矩形估计、旋转后的圆柱体及曲面等对象，通过滑块和输入框交互调整参数。④ 适合高中至大学阶段的微积分教学，帮助学生直观理解积分求体积的几何意义和数值逼近思想。

GeoGebra 命令及参数

  1. Textfield(f) → inputBox1
  2. Textfield(upperlimit) → inputBox2
  3. Textfield(lowerlimit) → inputBox3
  4. Integral(f, ll, ul) → AreabyDefiniteIntegral
  5. Sequence(Polygon[(dx, 0), (dx, f(dx)), (dx + (ul - ll) / number, f(dx)), (dx + (ul - ll) / number, 0)], dx, ll, ul - (ul - ll) / number, (ul - ll) / number) → LeftRectanglesEstimate
  6. Sum(LeftRectanglesEstimate) → AreaLeftRectangleEstimate
  7. Sequence(Polygon[(dx, 0), (dx, f(dx + (ul - ll) / number)), (dx + (ul - ll) / number, f(dx + (ul - ll) / number)), (dx + (ul - ll) / number, 0)], dx, ll, ul - (ul - ll) / number, (ul - ll) / number) → RightRectanglesEstimate
  8. Sum(RightRectanglesEstimate) → AreaRightRectangleEstimate
  9. Sequence(x = dx, dx, ll, ul, (ul - ll) / number) → VerticalLinesWithinLimits
  10. Sequence((dx, f(dx)), dx, ll, ul, (ul - ll) / number) → PointsOnCurvefWithinLimits
  11. Surface(fprime, θ, xAxis) → CurvedBoundary
  12. CircumcircleSector((ll, f(ll), 0), (ll, (f(ll) * cos(θ / 3)), (f(ll) * sin(θ / 3))), (ll, (f(ll) * cos(θ)), (f(ll) * sin(θ)))) → LLDisc
  13. CircumcircleSector((ul, f(ul), 0), (ul, (f(ul) * cos(θ / 3)), (f(ul) * sin(θ / 3))), (ul, (f(ul) * cos(θ)), (f(ul) * sin(θ)))) → ULDisc
  14. Sequence(Polygon[(dx, 0, 0), (dx, (f(dx) * cos(θ)), (f(dx) * sin(θ))), (dx + (ul - ll) / number, (f(dx) * cos(θ)), (f(dx) * sin(θ))), (dx + (ul - ll) / number, 0, 0)], dx, ll, ul - (ul - ll) / number, (ul - ll) / number) → LeftRectangleRotate
  15. Sequence(CircumcircleSector[(dx, f(dx), 0), (dx, (f(dx) * cos(θ / 3)), (f(dx) * sin(θ / 3))), (dx, (f(dx) * cos(θ)), (f(dx) * sin(θ)))], dx, ll, ul - (ul - ll) / number, (ul - ll) / number) → LeftCylinderLeftSector
  16. Sequence(CircumcircleSector[(dx + (ul - ll) / number, f(dx), 0), (dx + (ul - ll) / number, (f(dx) * cos(θ / 3)), (f(dx) * sin(θ / 3))), (dx + (ul - ll) / number, (f(dx) * cos(θ)), (f(dx) * sin(θ)))], dx, ll, ul - (ul - ll) / number, (ul - ll) / number) → LeftCylinderRightSector
  17. Sequence(Surface[u, (f(dx) * cos(v)), (f(dx) * sin(v)), u, dx, dx + (ul - ll) / number, v, 0, θ], dx, ll, ul - (ul - ll) / number, (ul - ll) / number) → LeftCylinderCurvedSurface
  18. Sequence(Cylinder[(dx, 0, 0), (dx + (ul - ll) / number, 0, 0), f(dx + (ul - ll) / number)], dx, ll, ul - (ul - ll) / number, (ul - ll) / number) → RightCylinderEstimate
  19. Sequence(CircumcircleSector[(dx, f(dx + (ul - ll) / number), 0), (dx, (f(dx + (ul - ll) / number) * cos(θ / 3)), (f(dx + (ul - ll) / number) * sin(θ / 3))), (dx, (f(dx + (ul - ll) / number) * cos(θ)), (f(dx + (ul - ll) / number) * sin(θ)))], dx, ll, ul - (ul - ll) / number, (ul - ll) / number) → RightCylinderLeftSector
  20. Sequence(CircumcircleSector[(dx + (ul - ll) / number, f(dx + (ul - ll) / number), 0), (dx + (ul - ll) / number, (f(dx + (ul - ll) / number) * cos(θ / 3)), (f(dx + (ul - ll) / number) * sin(θ / 3))), (dx + (ul - ll) / number, (f(dx + (ul - ll) / number) * cos(θ)), (f(dx + (ul - ll) / number) * sin(θ)))], dx, ll, ul - (ul - ll) / number, (ul - ll) / number) → RightCylinderRightSector
  21. Sequence(Polygon[(dx, 0, 0), (dx, (f(dx + (ul - ll) / number) * cos(θ)), (f(dx + (ul - ll) / number) * sin(θ))), (dx + (ul - ll) / number, (f(dx + (ul - ll) / number) * cos(θ)), (f(dx + (ul - ll) / number) * sin(θ))), (dx + (ul - ll) / number, 0, 0)], dx, ll, ul - (ul - ll) / number, (ul - ll) / number) → RightRectangleRotate
  22. Sum(RightCylinderEstimate) → VolumeRightCylinderEstimate
  23. Sequence(Cylinder[(dx, 0, 0), (dx + (ul - ll) / number, 0, 0), f(dx)], dx, ll, ul - (ul - ll) / number, (ul - ll) / number) → LeftCylinderEstimate
  24. Sum(LeftCylinderEstimate) → VolumeLeftCylinderEstimate
  25. Integral(f(x)^(2)) → h_1
  26. Sequence(Surface[u, (f(dx + (ul - ll) / number) * cos(v)), (f(dx + (ul - ll) / number) * sin(v)), u, dx, dx + (ul - ll) / number, v, 0, θ], dx, ll, ul - (ul - ll) / number, (ul - ll) / number) → RightCylinderCurvedSurface
  27. Surface(u, (v * f(u)), 0, u, lowerlimit, upperlimit, v, 0, 1) → Areain3D

AI 绘图提示语参考

绘制一个函数曲线绕x轴旋转形成的旋转体体积的可视化图形，展示使用定积分计算体积的过程，并包含左右矩形估计。图形应包括输入框用于设置函数和积分上下限，显示定积分计算的区域，左右矩形估计的图形，旋转形成的圆柱体和曲面，以及左右圆柱体的体积估计。使用滑动条和输入框进行参数交互，整体呈现旋转体、积分区域、矩形估计和旋转体的3D可视化。
以上提示语为根据原始绘图命令整理而来，仅供参考。