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Integral Volume Y: Visualizing Volume of Revolution

Integral Volume Y: Visualizing Volume of Revolution
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2018-05-06 09:11
FunctionSolid GeometryCalculusIntegralSlider Animation
① This GeoGebra example dynamically demonstrates the computation of the volume of a solid of revolution obtained by rotating a function curve around the x-axis, using definite integrals and also showing left and right rectangle estimates. ② Core mathematical concepts include definite integrals, volume of revolution, Riemann sums, cylindrical shell method, and 3D visualization of surfaces and solids. ③ Construction highlights: commands such as Textfield, Integral, Sequence, Sum, Surface, and CircumcircleSector are used to create input boxes for limits, the definite integral area, left/right rectangle estimates, rotated cylinders, and curved surfaces, with sliders and input boxes for interactive parameter changes. ④ Suitable for high school to college calculus courses, helping students intuitively grasp the geometric meaning of integration for volume and numerical approximation methods.

GeoGebra Commands & Parameters

  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 Drawing Prompt Reference

Visualize the volume of a solid of revolution formed by rotating a function curve around the x-axis using definite integrals, and include left and right rectangle estimates. The plot should feature input boxes for the function and integral limits, display the area of the definite integral, and show the left and right rectangle estimates. Include the rotated cylinders and curved surfaces, along with volume estimates for the left and right cylinders. Use sliders and input boxes for interactive parameter changes, presenting the rotation body, integral area, rectangle estimates, and 3D visualization of the solid.
The prompt above was compiled from the original drawing commands and is for reference only.