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Shell Method for Solids of Revolution

Shell Method for Solids of Revolution
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2015-09-13 13:21
FunctionTrigonometrySolid GeometryCalculusIntegral
This example demonstrates the calculation of solid of revolution volume using the shell method. It involves core math concepts like definite integral, rotation, solid geometry volume, quadratic functions, natural numbers, trigonometry, etc. Construction highlights the use of GeoGebra commands/objects such as Point, CurveCartesian, Integral, Segment, Line, Surface, Sequence, with elements like yMax, yMin, xMin, xMax, UpperFunction, ShellBar, etc., to dynamically visualize volume integration. Suitable for high school and above students, and calculus education purposes.

GeoGebra Commands & Parameters

  1. Point(y = 0) → R
  2. Point(xAxis) → RightEnd
  3. Point(xAxis) → LeftEnd
  4. CurveCartesian(t, (0.8 * yMin) + (0.2 * yMax), t, x(LeftEnd), x(RightEnd)) → ShellBar
  5. Point(ShellBar) → ShellPoint
  6. CurveCartesian(t, UpperFunction(t), t, x(LeftEnd), x(RightEnd)) → UpperCurve
  7. Textfield(UpperFunction) → UpperFunctionBox
  8. Integral(ShellArea, x(LeftEnd), x(ShellPoint)) → VolumeToDisk
  9. Integral(ShellArea, x(LeftEnd), x(RightEnd)) → FullVolume
  10. Textfield(xMin) → xMinBox
  11. Textfield(xMax) → xMaxBox
  12. Textfield(yMin) → yMinBox
  13. Textfield(yMax) → yMaxtBox
  14. CurveCartesian(t, LowerFunction(t), t, x(LeftEnd), x(RightEnd)) → LowerCurve
  15. Segment((x(ShellPoint), LowerFunction(x(ShellPoint))), (x(ShellPoint), UpperFunction(x(ShellPoint)))) → LineToCurve
  16. Textfield(LowerFunction) → LowerFunctionBox
  17. Line(R, R + (0, 1)) → AxisOfRotation
  18. Surface(x(R) + (cos(t) * (x(ShellPoint) - x(R))), (s * UpperFunction(x(ShellPoint))) + ((1 - s) * LowerFunction(x(ShellPoint))), (sin(t) * (x(ShellPoint) - x(R))), s, 0, 1, t, 0, 6.28319) → IntegrationDisk
  19. Surface(x(R) + (cos(t) * (s - x(R))), UpperFunction(s), (sin(t) * (s - x(R))), s, x(LeftEnd), x(RightEnd), t, 6.2832 - RotationAngleShown, 6.2832) → SurfaceRev
  20. Surface(x(R) + (cos(t) * (s - x(R))), LowerFunction(s), (sin(t) * (s - x(R))), s, x(LeftEnd), x(RightEnd), t, 6.2832 - RotationAngleShown, 6.2832) → SurfaceRevInner
  21. Sequence(CurveCartesian[x(R) + (cos((((n + 0.5) * 2) * π / 12)) * (t - x(R))), UpperFunction(t), (sin((((n + 0.5) * 2) * π / 12)) * (t - x(R))), t, x(LeftEnd), x(RightEnd)], n, 0, 11) → WireFrameRibs
  22. Sequence(CurveCartesian[x(R) + (cos((((n + 0.5) * 2) * π / 12)) * (t - x(R))), LowerFunction(t), (sin((((n + 0.5) * 2) * π / 12)) * (t - x(R))), t, x(LeftEnd), x(RightEnd)], n, 0, 11) → WireFrameRibsInner