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Volume by Washers by Mike May (3)

Volume by Washers by Mike May (3)
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2016-03-20 09:48
CalculusIntegralGeometric ConstructionSlider Animation
In collections AP Calculus volume problems Calculus - Applications of Integrals - Areas and Volumes
This example demonstrates the calculation of the volume of a solid of revolution from the 2007 AP Exam Problem 1b. Key mathematical concepts include integration and volume of solids of revolution. It is constructed using GeoGebra commands such as Sequence, Polygon, Rotate, and Integral. Suitable for high school math education and learning applications of integration in calculus.

GeoGebra Commands & Parameters

  1. Sequence(Polygon[(-2 + (k * h), f(-2 + ((k + p) * h))), (-2 + ((k + 1) * h), f(-2 + ((k + p) * h))), (-2 + ((k + 1) * h), g(-2 + ((k + p) * h))), (-2 + (k * h), g(-2 + ((k + p) * h)))], k, 0, m - 1) → Rectangles
  2. Polygon((-2 + (DK * h), f(-2 + ((DK + p) * h))), (-2 + ((DK + 1) * h), f(-2 + ((DK + p) * h))), (-2 + ((DK + 1) * h), g(-2 + ((DK + p) * h))), (-2 + (DK * h), g(-2 + ((DK + p) * h)))) → Rectangle1, b, c, d, e
  3. Rotate(Rectangle1, α, xAxis) → Rectangle1'
  4. Integral(A, -2, 2) → VolInteg
  5. RectangleSum(A, -2, -2 + (m * h), m, p) → WS
  6. Surface(u, (f(u) * cos(β)), (f(u) * sin(β)), u, -2, 2, β, 0, (2 * π)) → Inner
  7. Surface(u, (g(u) * cos(β)), (g(u) * sin(β)), u, -2, 2, β, 0, (2 * π)) → Outer
  8. Surface(-2 + (DK * h), (u * cos(β)), (u * sin(β)), u, f(-2 + ((DK + p) * h)), g(-2 + ((DK + p) * h)), β, 0, α) → SideLeft1
  9. Surface(-2 + ((DK + 1) * h), (u * cos(β)), (u * sin(β)), u, f(-2 + ((DK + p) * h)), g(-2 + ((DK + p) * h)), β, 0, α) → SideRight1
  10. Surface(-2 + u, (f(-2 + ((DK + p) * h)) * cos(β)), (f(-2 + ((DK + p) * h)) * sin(β)), u, (DK * h), ((DK + 1) * h), β, 0, α) → Inner1
  11. Surface(-2 + u, (g(-2 + ((DK + p) * h)) * cos(β)), (g(-2 + ((DK + p) * h)) * sin(β)), u, (DK * h), ((DK + 1) * h), β, 0, α) → Outer1
  12. Rotate(g_1, α, xAxis) → i
  13. Rotate(f_1, α, xAxis) → j
  14. Rotate(Rectangles, α, xAxis) → Rectangles'
  15. Sequence(Surface[-2 + u, (f(-2 + ((k + p) * h)) * cos(β)), (f(-2 + ((k + p) * h)) * sin(β)), u, (k * h), ((k + 1) * h), β, 0, α], k, 0, m - 1) → Inners
  16. Sequence(Surface[-2 + u, (g(-2 + ((k + p) * h)) * cos(β)), (g(-2 + ((k + p) * h)) * sin(β)), u, (k * h), ((k + 1) * h), β, 0, α], k, 0, m - 1) → Outers
  17. Sequence(Surface[-2 + (k * h), (u * cos(β)), (u * sin(β)), u, f(-2 + ((k + p) * h)), g(-2 + ((k + p) * h)), β, 0, α], k, 0, m - 1) → SidesLeft
  18. Sequence(Surface[-2 + ((k + 1) * h), (u * cos(β)), (u * sin(β)), u, f(-2 + ((k + p) * h)), g(-2 + ((k + p) * h)), β, 0, α], k, 0, m - 1) → SidesRight