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Region Bounded by Two Quadratics Rotated about x=2

Region Bounded by Two Quadratics Rotated about x=2
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2015-07-02 12:16
FunctionCoordinate GeometrySolid GeometryCalculus
In collections Volumes in Calculus Volumes in Calculus
This example demonstrates the region bounded by two quadratic functions rotated about the x=2 axis. Key mathematical concepts include quadratic functions, rotation, integration, and solid geometry. Using GeoGebra commands such as Rotate, Polygon, Vector, and Integral, the region can be constructed. Suitable for high school mathematics education, particularly in coordinate geometry and calculus.

GeoGebra Commands & Parameters

  1. Rotate(f_1, α, a) → f_3
  2. Rotate(g_1, α, a) → g_3
  3. Polygon((((m - 1) * h) - 2, g(((m - 1 + p) * h) - 2)), ((m * h) - 2, g(((m - 1 + p) * h) - 2)), ((m * h) - 2, f(((m - 1 + p) * h) - 2)), (((m - 1) * h) - 2, f(((m - 1 + p) * h) - 2))) → Rectangle1, i, j, k, l
  4. Rotate(Rectangle1, α, a) → Rectangle1'
  5. Vector((-2 + ((DK + p) * h), g(-2 + ((DK + p) * h))), (-2 + ((DK + p) * h), f(-2 + ((DK + p) * h)))) → U
  6. Vector((2, g(-2 + ((DK + p) * h))), (-2 + ((DK + p) * h), g(-2 + ((DK + p) * h)))) → V
  7. Rotate(V, α, a) → R'
  8. Rotate(A, α, a) → A'
  9. Rotate(B, α, a) → B'
  10. Vector(A', B') → H'
  11. Integral(((f(x) - g(x)) * (2 - x)), -2, 1) → Vol
  12. Surface(((2 - u) * (-cos(β))) + 2, g(u), ((2 - u) * sin(β)), u, -2, 1, β, 0, 6.28319) → SurfaceFront
  13. Surface(((2 - u) * (-cos(β))) + 2, f(u), ((2 - u) * sin(β)), u, -2, 1, β, 0, 6.28319) → SurfaceBack
  14. If(togS ≟ 1, "Hide surface", "Show surface") → ShowSurface
  15. Sequence(Polygon[(-2 + (k * h), g(-2 + ((k + p) * h))), (-2 + ((k + 1) * h), g(-2 + ((k + p) * h))), (-2 + ((k + 1) * h), f(-2 + ((k + p) * h))), (-2 + (k * h), f(-2 + ((k + p) * h)))], k, 0, m - 1) → Rectangles
  16. Rotate(Rectangles, α, a) → Rectangles'
  17. If(togSH ≟ 1, "Hide Shells", "Show Shells") → ShowShell
  18. If(togR ≟ 0, "Show one shell", "Show all shells") → ShowOneShell
  19. If(togV ≟ 1, "Hide volume integral", "Show volume integral") → ShowVolume
  20. Sequence(Surface[2 - ((4 - (k * h)) * cos(β)), (v * g(-2 + ((k + p) * h))) + ((1 - v) * f(-2 + ((k + p) * h))), ((4 - (k * h)) * sin(β)), v, 0, 1, β, 0, α], k, 0, m - 1) → Outers
  21. Sequence(Surface[2 - ((4 - ((k + 1) * h)) * cos(β)), (v * g(-2 + ((k + p) * h))) + ((1 - v) * f(-2 + ((k + p) * h))), ((4 - ((k + 1) * h)) * sin(β)), v, 0, 1, β, 0, α], k, 0, m - 1) → Inners
  22. Sequence(Surface[2 - ((4 - ((k + u) * h)) * cos(β)), g(-2 + ((k + p) * h)), ((4 - ((k + u) * h)) * sin(β)), u, 0, 1, β, 0, α], k, 0, m - 1) → Fronts
  23. Sequence(Surface[2 - ((4 - ((k + u) * h)) * cos(β)), f(-2 + ((k + p) * h)), ((4 - ((k + u) * h)) * sin(β)), u, 0, 1, β, 0, α], k, 0, m - 1) → Backs
  24. RectangleSum((((2 * π) * R(x)) * H(x)), -2, -2 + (m * h), m, p) → VolShells
  25. Surface(((4 - u) * (-cos(β))) + 2, g(((m - 1 + p) * h) - 2), ((4 - u) * sin(β)), u, ((m - 1) * h), (m * h), β, 0, α) → Bottom1
  26. Surface(((4 - (m * h)) * (-cos(β))) + 2, v, ((4 - (m * h)) * sin(β)), v, g(((m - 1 + p) * h) - 2), f(((m - 1 + p) * h) - 2), β, 0, α) → InnerSurface1
  27. Surface(((4 - ((m - 1) * h)) * (-cos(β))) + 2, v, ((4 - ((m - 1) * h)) * sin(β)), v, g(((m - 1 + p) * h) - 2), f(((m - 1 + p) * h) - 2), β, 0, α) → OuterSurface1
  28. Surface(((4 - u) * (-cos(β))) + 2, f(((m - 1 + p) * h) - 2), ((4 - u) * sin(β)), u, ((m - 1) * h), (m * h), β, 0, α) → Top1

AI Drawing Prompt Reference

Create a three‑dimensional solid of revolution formed by rotating the region between two quadratic curves around the vertical line x = 2. Show the outer and inner curved surfaces, as well as the front and back faces, and include top and bottom caps. Illustrate the solid with a series of cylindrical shells that approximate its volume, giving a clear view of the shape’s structure and depth. Use a slightly transparent or shaded appearance to highlight the layered surfaces and the space between the curves.
The prompt above was compiled from the original drawing commands and is for reference only.