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Volume of a solid of revolution (shell method)

Volume of a solid of revolution (shell method)
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2020-01-28 10:18
FunctionSolid GeometryCalculusIntegral
This example demonstrates the calculation of the volume of a solid of revolution using the shell method. Key math concepts include integration, volume of solids, and function graphing. It is constructed using commands like Line, Segment, CurveCartesian, Sequence, Textfield, and involves objects such as e, aa, bb, n, ff, st, yDir, ss, f, Qss, Pss, PQ, gg, fInRotation, g, gInRotation, st2, lateralSurfAA, lateralSurfBB, etc. Suitable for high school and above students learning calculus and geometric transformations.

GeoGebra Commands & Parameters

  1. Line((0, 0, 0), (0, 1, 0)) → yDir
  2. Segment(Pss, Qss) → PQ
  3. CurveCartesian((t * cos(ss)), ff(t), (t * sin(ss)), t, aa, bb) → fInRotation
  4. CurveCartesian((t * cos(ss)), gg(t), (t * sin(ss)), t, aa, bb) → gInRotation
  5. Sequence(Segment[((aa * cos(k)), gg(aa), (aa * sin(k))), ((aa * cos(k)), ff(aa), (aa * sin(k)))], k, 0, (2 * π), st2) → lateralSurfAA
  6. Sequence(Segment[((bb * cos(k)), gg(bb), (bb * sin(k))), ((bb * cos(k)), ff(bb), (bb * sin(k)))], k, 0, (2 * π), st2) → lateralSurfBB
  7. Sequence(Segment[(t, gg(t), 0), (t, ff(t), 0)], t, aa, bb, st3) → region
  8. Sequence(Segment[((t * cos(ss)), gg(t), (t * sin(ss))), ((t * cos(ss)), ff(t), (t * sin(ss)))], t, aa, bb, st3) → regionInRotation
  9. Sequence(Circle[(0, gg(aa + (k * st)), 0), aa + (k * st), yDir], k, 0, n, st1) → lowerCircles
  10. Sequence(Circle[(0, ff(aa + (k * st)), 0), aa + (k * st), yDir], k, 0, n, st1) → upperCircles
  11. Textfield(ff) → InputBox1
  12. Textfield(gg) → InputBox2
  13. Textfield(aa) → InputBox3
  14. Textfield(bb) → InputBox4
  15. Textfield(st1) → InputBox5
  16. Textfield(st2) → InputBox6
  17. Sequence(Segment[(((tt + dx) * cos(k)), gg(tt), ((tt + dx) * sin(k))), (((tt + dx) * cos(k)), ff(tt), ((tt + dx) * sin(k)))], k, 0, (2 * π), stshell) → outerShell
  18. Sequence(Segment[((tt * cos(k)), gg(tt), (tt * sin(k))), ((tt * cos(k)), ff(tt), (tt * sin(k)))], k, 0, (2 * π), stshell) → innerShell
  19. Sequence(Circle[(0, ff(tt), 0), k, yDir], k, tt, tt + dx, stshell) → upperShellBase
  20. Textfield(stshell) → InputBox8
  21. Textfield(tt) → InputBox9
  22. Textfield(gg) → InputBox7