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Rotation of a Function Around the x‑Axis (Volume)

Rotation of a Function Around the x‑Axis (Volume)
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2012-12-05 04:16
FunctionSolid GeometryCalculusIntegralTransformation
① This example demonstrates the solid generated by rotating an arbitrary function around the x‑axis and shows its volume in real time. ② Core mathematical concepts include definite integrals for volume, solids of revolution, and the interplay between function graphs and solid geometry. ③ Construction highlights: the function is defined with Function; slices are created via Sequence; Vector, OrthogonalLine, Translate, Mirror, Segment, Point, Line, Intersect, and Textfield build the 3D rotation view and interactive input. ④ Suitable for high‑school calculus, university multivariable calculus lectures, and self‑study of volume integration visualization.

GeoGebra Commands & Parameters

  1. Vector(B_1, X) → u_1
  2. Vector(B_1, Z) → w
  3. OrthogonalLine(View3D, yAxis) → a
  4. OrthogonalLine(View3D, xAxis) → b
  5. Vector(B_1, Y) → v
  6. Translate(O, Vector[u * u_1]) → O'_1
  7. Translate(O, Vector[u * v]) → O'_2
  8. Vector(O, O'_2) → z
  9. Vector(O, O'_1) → d
  10. Translate(O, Vector[u * w]) → O'_3
  11. Vector(O, O'_3) → e
  12. Mirror(O'_1, O) → G
  13. Mirror(O, O'_2) → O'_4
  14. Translate(O, Vector[-u * w]) → O'_5
  15. Translate(O, Vector[-u * v]) → O'
  16. Segment(O, O'_5) → i
  17. Segment(O, O') → j
  18. Point(xAxis) → H
  19. Line(H, yAxis) → l
  20. Point(xAxis) → I
  21. Sequence(O + i * v, i, -3, 3) → liste5
  22. Sequence(O + i * w, i, -3, 3) → liste6
  23. Sequence(O + i * u_1, i, -3, 3) → liste7
  24. Intersect(xAxis, yAxis) → J
  25. Point(yAxis) → K
  26. Point(yAxis) → L
  27. Vector(K, L) → q
  28. Point(xAxis) → M
  29. Vector(I, M) → r
  30. Sequence((x(J) + i, y(J)), i, -3, 3) → liste8
  31. Sequence((x(J), y(J) + i), i, -3, 3) → liste9
  32. Function(f, Min, Max) → p
  33. Mirror(O'_1, O) → N
  34. Segment(N, O) → n_1
  35. Sequence(CurveCartesian[x(O + t * u_1 + f(t) * cos(s) * v - f(t) * sin(s) * w), y(O + t * u_1 + f(t) * cos(s) * v - f(t) * sin(s) * w), t, Min, Max], s, 0, 2 * 3.141592653589793 * Rot / 24, 3.141592653589793 / dx) → liste10
  36. Sequence(CurveCartesian[x(O + t * u_1 + f(t) * cos(s) * v - f(t) * sin(s) * w), y(O + t * u_1 + f(t) * cos(s) * v - f(t) * sin(s) * w), s, 0, 2 * 3.141592653589793 * m / 24], t, Min, Max, (Max - Min) / dx) → liste11
  37. Textfield(f) → ChampTexte1

AI Drawing Prompt Reference

Draw a 3D visualization of a solid of revolution created by rotating an arbitrary function around the x‑axis, showing the coordinate axes, the function curve in the xy‑plane, the solid built from a stack of circular disks, a text field to edit the function, a slider to adjust the rotation angle or number of slices, and a live display of the computed volume.
The prompt above was compiled from the original drawing commands and is for reference only.