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Volume of Revolution--Washers

Volume of Revolution--Washers
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2016-08-12 07:17
Solid GeometryCalculusIntegral
This example demonstrates the calculation of the volume of a solid of revolution using the washer method. Key math concepts include integral, limit, and function graphing. Construction highlights the use of GeoGebra commands and objects like Sequence, Polygon, Rotate, Integral, Surface, and RectangleSum to model washers, with slider animations showing volume changes. Suitable for high school and above students learning calculus-based volume calculations.

GeoGebra Commands & Parameters

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