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Volume Generated by Rotating Region about x=2

Volume Generated by Rotating Region about x=2
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2015-07-02 12:29
Solid GeometryCalculusGeometric Construction
In collections Volumes in Calculus Volumes in Calculus
This example demonstrates how to generate a volume by rotating a planar region around the line x=2. Key mathematical concepts include calculus volume integrals and solid geometry. Construction involves GeoGebra commands like Surface, Polygon, Integral, and Sum. Suitable for high school students or college freshmen learning calculus and solid geometry.

GeoGebra Commands & Parameters

  1. Sequence(Cylinder[(2, g(((k + p) * h)), 0), (2, f(((k + p) * h)), 0), 2 - ((k + 1) * h)], k, 0, m - 1) → CI
  2. Sequence(If[Defined[Element[CI, k]], Volume[Element[CI, k]], 0], k, 1, m) → VCI
  3. Sequence(Cylinder[(2, g(((k + p) * h)), 0), (2, f(((k + p) * h)), 0), 2 - (k * h)], k, 0, m - 1) → CO
  4. Sequence(If[Defined[Element[CO, k]], Volume[Element[CO, k]], 0], k, 1, m) → VCO
  5. Rotate(f_1, α, a) → f_3
  6. Rotate(g_1, α, a) → g_3
  7. Max({0, m - 1}) → DK
  8. Surface(((2 - u) * (-cos(β))) + 2, g(((DK + p) * h)), ((2 - u) * sin(β)), u, (DK * h), ((DK + 1) * h), β, 0, α) → Bottom1
  9. Surface(((2 - ((DK + 1) * h)) * (-cos(β))) + 2, v, ((2 - ((DK + 1) * h)) * sin(β)), v, g(((DK + p) * h)), f(((DK + p) * h)), β, 0, α) → InnerSurface1
  10. Surface(((2 - (DK * h)) * (-cos(β))) + 2, v, ((2 - (DK * h)) * sin(β)), v, g(((DK + p) * h)), f(((DK + p) * h)), β, 0, α) → OuterSurface1
  11. Surface(((2 - u) * (-cos(β))) + 2, f(((DK + p) * h)), ((2 - u) * sin(β)), u, (DK * h), ((DK + 1) * h), β, 0, α) → Top1
  12. Polygon({((DK * h), g(((DK + p) * h))), (((DK + 1) * h), g(((DK + p) * h))), (((DK + 1) * h), f(((DK + p) * h))), ((DK * h), f(((DK + p) * h)))}) → Rectangle1
  13. Rotate(Rectangle1, α, a) → RotatedRectangle1
  14. Vector((2, g(((DK + p) * h)), 0), (((DK + p) * h), g(((DK + p) * h)))) → V
  15. Rotate(V, α, a) → R'
  16. Rotate(A, α, a) → A'
  17. Rotate(B, α, a) → B'
  18. Vector(A, B) → U
  19. Vector(A', B') → H'
  20. Integral(((2 - x) * (f(x) - g(x))), 0, 2) → a_1
  21. Sum(VCO - VCI) → VolShellSum
  22. Sequence(Polygon[((k * h), g(((k + p) * h))), (((k + 1) * h), g(((k + p) * h))), (((k + 1) * h), f(((k + p) * h))), ((k * h), f(((k + p) * h)))], k, 0, m - 1) → Rectangles
  23. If(togV ≟ 1, "Hide integral & shell approx", "Show integral & shell approx") → ShowVolumeIntegral
  24. Rotate(Rectangles, α, a) → Rectangles'
  25. If(togR ≟ 1, "Show One Shell", "Show All Shells") → ShowOneShell
  26. Surface(((2 - u / 2) * cos(β)) + 2, u, ((2 - u / 2) * sin(β)), u, 0, 4, β, 0, (2 * π)) → Outer
  27. Surface(((2 - sqrt(u)) * cos(β)) + 2, u, ((2 - sqrt(u)) * sin(β)), u, 0, 4, β, 0, (2 * π)) → Inner
  28. If(togS ≟ 1, "Hide surface", "Show surface") → ShowSurface
  29. Sequence(Surface[2 - ((2 - (k * h)) * cos(β)), (v * f(((k + p) * h))) + ((1 - v) * g(((k + p) * h))), ((2 - (k * h)) * sin(β)), v, 0, 1, β, 0, α], k, 0, m - 1) → Outers
  30. Sequence(Surface[2 - ((2 - ((k + 1) * h)) * cos(β)), (v * f(((k + p) * h))) + ((1 - v) * g(((k + p) * h))), ((2 - ((k + 1) * h)) * sin(β)), v, 0, 1, β, 0, α], k, 0, m - 1) → Inners
  31. Sequence(Surface[2 - ((2 - ((k + u) * h)) * cos(β)), g(((k + p) * h)), ((2 - ((k + u) * h)) * sin(β)), u, 0, 1, β, 0, α], k, 0, m - 1) → Fronts
  32. Sequence(Surface[2 - ((2 - ((k + u) * h)) * cos(β)), f(((k + p) * h)), ((2 - ((k + u) * h)) * sin(β)), u, 0, 1, β, 0, α], k, 0, m - 1) → Backs
  33. RectangleSum((((2 * π) * R(x)) * H(x)), 0, (m * h), m, p) → VolShells

AI Drawing Prompt Reference

Create a three‑dimensional illustration of the solid obtained by rotating the planar region bounded by an upper curve and a lower curve from x = 0 to x = 2 around the vertical line x = 2. Show the resulting hollow solid with its outer surface, inner surface, top and bottom faces. Include a series of cylindrical shells (or sliced washers) that approximate the volume, and depict the original rectangular region before rotation for reference.
The prompt above was compiled from the original drawing commands and is for reference only.