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Volume by Washers by Mike May

Volume by Washers by Mike May
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2016-03-20 02:44
CalculusIntegralGeometric Construction
In collections AP Calculus volume problems
This example demonstrates the volume calculation of a solid of revolution using the washer method. Key mathematical concepts include integration and volume of solids of revolution. Using GeoGebra commands such as Sequence, Polygon, Rotate, and Integral, the cross-sections of the solid are constructed, and the volume is calculated through integration. Suitable for high school calculus courses.

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