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Volumes with Washers Demonstration

Volumes with Washers Demonstration
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2019-04-04 09:29
FunctionSolid GeometryCalculusIntegralGeometric Construction
This example demonstrates the calculation of volume using the washer method for rotating solids. It involves core mathematical concepts such as integration, volume of solids of revolution, and function graphing. GeoGebra commands like Sequence, Polygon, Rotate, Integral, RectangleSum, and Surface are used in the construction. Suitable for high school and above students to learn and understand the application of calculus in geometric 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. Rotate(Rectangles, α, a) → Rectangles'
  8. RectangleSum(A, 0, (m * h), m, p) → WS
  9. Surface(u, -2 - ((-2 - g(u)) * cos(β)), ((-2 - g(u)) * sin(β)), u, 0, 2, β, 0, (2 * π)) → Inner
  10. Surface(u, -2 - ((-2 - g(((DK + p) * h))) * cos(β)), ((-2 - g(((DK + p) * h))) * sin(β)), u, (DK * h), ((DK + 1) * h), β, 0, α) → Inner1
  11. Surface(u, -2 - ((-2 - f(u)) * cos(β)), ((-2 - f(u)) * sin(β)), u, 0, 2, β, 0, (2 * π)) → Outer
  12. Surface(u, -2 - ((-2 - f(((DK + p) * h))) * cos(β)), ((-2 - f(((DK + p) * h))) * sin(β)), u, (DK * h), ((DK + 1) * h), β, 0, α) → Outer1
  13. Sequence(Surface[u, -2 - ((-2 - g(((k + p) * h))) * cos(β)), ((-2 - g(((k + p) * h))) * sin(β)), u, (k * h), ((k + 1) * h), β, 0, α], k, 0, m - 1) → Inners
  14. Sequence(Surface[u, -2 - ((-2 - f(((k + p) * h))) * cos(β)), ((-2 - f(((k + p) * h))) * sin(β)), u, (k * h), ((k + 1) * h), β, 0, α], k, 0, m - 1) → Outers
  15. Sequence(Surface[(k * h), -2 - ((-2 - u) * cos(β)), ((-2 - u) * sin(β)), u, g(((k + p) * h)), f(((k + p) * h)), β, 0, α], k, 0, m - 1) → SidesLeft
  16. Sequence(Surface[((k + 1) * h), -2 - ((-2 - u) * cos(β)), ((-2 - u) * sin(β)), u, g(((k + p) * h)), f(((k + p) * h)), β, 0, α], k, 0, m - 1) → SidesRight
  17. Surface((DK * h), -2 - ((-2 - u) * cos(β)), ((-2 - u) * sin(β)), u, g(((DK + p) * h)), f(((DK + p) * h)), β, 0, α) → SideLeft1
  18. Surface((DK * h), -2 - ((-2 - u) * cos(β)), ((-2 - u) * sin(β)), u, g(((DK + p) * h)), f(((DK + p) * h)), β, 0, α) → SideRight1