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Integral Volume Demonstration

Integral Volume Demonstration
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2018-05-05 11:43
FunctionSolid GeometryCalculusIntegralGeometric Construction
This example demonstrates the calculation of the volume of a solid of revolution formed by rotating a curve around the X-axis using integration. Key math concepts include integration, solid of revolution, function graphs, and limits. It utilizes GeoGebra commands like Integral, Sequence, Sum, Surface, and various objects such as f, limits, and geometric constructs like LLDisc, ULDisc, and cylinders. Suitable for high school and above, particularly for understanding the application of integration in volume calculations in calculus and solid geometry.

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