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积分体积演示

积分体积演示
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2018-05-05 11:43
函数立体几何微积分积分几何作图
①本范例演示通过积分计算旋转体的体积，以X轴为旋转轴的曲线旋转形成的立体图形；②涉及的核心数学概念包括积分、旋转体体积、函数图像、极限等；③作图要点——从下方『构造摘要』提炼用到了哪些 GeoGebra 命令/对象：Textfield, Integral, Sequence, Sum, Surface, CircumcircleSector, f, upperlimit, lowerlimit, number, b, a, inputBox1, inputBox2, inputBox3, AreabyDefiniteIntegral, LeftRectanglesEstimate, RightRectanglesEstimate, VerticalLinesWithinLimits, PointsOnCurvefWithinLimits, h, θ, fprime, CurvedBoundary, LLDisc, ULDisc, LeftRectangleRotate, LeftCylinderLeftSector, LeftCylinderRightSector, LeftCylinderCurvedSurface, RightCylinderEstimate, RightCylinderLeftSector, RightCylinderRightSector, RightRectangleRotate, VolumeRightCylinderEstimate, LeftCylinderEstimate, VolumeLeftCylinderEstimate；④适合高中及以上学生学习微积分和立体几何，用于理解积分在体积计算中的应用。

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