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Rotation of a region around y-axis

Rotation of a region around y-axis
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2020-01-27 15:24
FunctionSolid GeometryCalculusIntegralTransformation
This example demonstrates the rotation of a planar region around the y-axis to form a solid of revolution. It involves core mathematical concepts such as rotation, parabola, sphere, median line, integral calculus, and angles. The construction utilizes GeoGebra commands and objects like Line, Segment, CurveCartesian, Sequence, Textfield, and entities like aa, bb, n, st, yDir, ss, ff, f, Qss, Pss, PQ, gg, fInRotation, g, gInRotation, st2, lateralSurfAA, lateralSurfBB, st3, region, regionInRotation, st1, lowerCircles, upperCircles, etc. It is suitable for high school and above, particularly for understanding volume calculation of solids of revolution and geometric transformations in integral calculus and solid geometry.

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