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Volume by Washers using Disk Method

Volume by Washers using Disk Method
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2016-03-20 05:49
FunctionSolid GeometryCalculusIntegral
This example demonstrates how to calculate the volume of a solid of revolution using the washer method. The core mathematical concepts involved are definite integrals and volume calculation of rotated solids. Construction highlights include the use of GeoGebra commands and objects such as Sequence, Rotate, Integral, If, Surface, Polygon, RectangleSum to build the geometry and compute the volume. Suitable for high school and introductory college-level calculus and solid geometry students.

GeoGebra Commands & Parameters

  1. Sequence(Polygon[((k * hx), g(((k + p) * hx))), (((k + 1) * hx), g(((k + p) * hx))), (((k + 1) * hx), f(((k + p) * hx))), ((k * hx), f(((k + p) * hx)))], k, 0, m - 1) → VRectangles
  2. Rotate(B, α, a) → B'
  3. Integral((((2 * π) * (1 + x)) * (f(x) - g(x))), 0, 9) → VolShellIntegral
  4. Integral((π * ((1 + gI(y))^(2) - (1 + fI(y))^(2))), 0, 3) → VolWasherIntegral
  5. Sequence(Polygon[(gI(((k + p) * hy)), (k * hy)), (fI(((k + p) * hy)), (k * hy)), (fI(((k + p) * hy)), ((k + 1) * hy)), (gI(((k + p) * hy)), ((k + 1) * hy))], k, 0, m - 1) → HRectangles
  6. Rotate(HRectangles, (-α), a) → HRectangles'
  7. Rotate(VRectangles, (-α), a) → VRectangles'
  8. If(togSurf ≟ 1, "Hide surface", "Show surface") → ShowSurface
  9. Surface(-1 + ((1 + gI(u)) * cos(β)), u, ((1 + gI(u)) * sin(β)), u, 0, 3, β, 0, (2 * π)) → Outer
  10. Surface(-1 + ((1 + fI(u)) * cos(β)), u, ((1 + fI(u)) * sin(β)), u, 0, 3, β, 0, (2 * π)) → Inner
  11. If(togSH ≟ 1, "Use washers", "Use shells") → ShellWasher
  12. Surface(-1 + ((1 + u) * cos(β)), f(((m - 1 + p) * hx)), ((1 + u) * sin(β)), u, ((m - 1) * hx), (m * hx), β, 0, α) → ShellsBack1
  13. Surface(-1 + ((1 + u) * cos(β)), (g(m - 1 + p) * hx), ((1 + u) * sin(β)), u, ((m - 1) * hx), (m * hx), β, 0, α) → ShellsFront1
  14. Surface(-1 + ((1 + ((m - 1) * hx)) * cos(β)), v, ((1 + ((m - 1) * hx)) * sin(β)), v, g(((m - 1 + p) * hx)), f(((m - 1 + p) * hx)), β, 0, α) → ShellsInner1
  15. Surface(-1 + ((1 + (m * hx)) * cos(β)), v, ((1 + (m * hx)) * sin(β)), v, g(((m - 1 + p) * hx)), f(((m - 1 + p) * hx)), β, 0, α) → ShellsOuter1
  16. Polygon((gI(((m - 1 + p) * hy)), ((m - 1) * hy)), (fI(((m - 1 + p) * hy)), ((m - 1) * hy)), (fI(((m - 1 + p) * hy)), (m * hy)), (gI(((m - 1 + p) * hy)), (m * hy))) → RectWash, b, c, d, e
  17. Polygon((((m - 1) * hx), g(((m - 1 + p) * hx))), (((m - 1) * hx), f(((m - 1 + p) * hx))), ((m * hx), f(((m - 1 + p) * hx))), ((m * hx), g(((m - 1 + p) * hx)))) → RectShell, h, i, j, k
  18. Surface(-1 + ((1 + fI(((m - 1 + p) * hy))) * cos(β)), v, ((1 + fI(((m - 1 + p) * hy))) * sin(β)), v, ((m - 1) * hy), (m * hy), β, 0, α) → WasherInner1
  19. Surface(-1 + ((1 + gI(((m - 1 + p) * hy))) * cos(β)), v, ((1 + gI(((m - 1 + p) * hy))) * sin(β)), v, ((m - 1) * hy), (m * hy), β, 0, α) → WashOuter1
  20. Surface(-1 + ((1 + u) * cos(β)), ((m - 1) * hy), ((1 + u) * sin(β)), u, fI(((m - 1 + p) * hy)), gI(((m - 1 + p) * hy)), β, 0, α) → WashFront1
  21. Surface(-1 + ((1 + u) * cos(β)), (m * hy), ((1 + u) * sin(β)), u, fI(((m - 1 + p) * hy)), gI(((m - 1 + p) * hy)), β, 0, α) → WashBack1
  22. RectangleSum((((2 * π) * (1 + x)) * (sqrt(x) - x / 3)), 0, (m * hx), m, p) → VolShells
  23. RectangleSum((π * ((1 + (3 * y))^(2) - (1 + y^(2))^(2))), 0, (m * hy), m, p) → VolWashers
  24. Sequence(Surface[-1 + ((1 + u) * cos(β)), ((k + 1) * hy), ((1 + u) * sin(β)), u, fI(((k + p) * hy)), gI(((k + p) * hy)), β, 0, α], k, 0, m - 1) → WashersBack
  25. Sequence(Surface[-1 + ((1 + fI(((k + p) * hy))) * cos(β)), ((k + v) * hy), ((1 + fI(((k + p) * hy))) * sin(β)), v, 0, 1, β, 0, α], k, 0, m - 1) → WashInners
  26. Sequence(Surface[-1 + ((1 + (u * fI(((k + p) * hy))) + ((1 - u) * gI(((k + p) * hy)))) * cos(β)), ((k + u) * hy), ((1 + (u * fI(((k + p) * hy))) + ((1 - u) * gI(((k + p) * hy)))) * sin(β)), u, 0, 1, β, 0, α], k, 0, m - 1) → WasherFronts
  27. Sequence(Surface[-1 + ((1 + gI(((k + p) * hy))) * cos(β)), ((k + v) * hy), ((1 + gI(((k + p) * hy))) * sin(β)), v, 0, 1, β, 0, α], k, 0, m - 1) → WasherOuters
  28. Sequence(Surface[-1 + ((1 + u) * cos(β)), f(((k + p) * hx)), ((1 + u) * sin(β)), u, (k * hx), ((k + 1) * hx), β, 0, α], k, 0, m - 1) → ShellsBack
  29. Sequence(Surface[-1 + ((1 + u) * cos(β)), f(((k + p) * hx)), ((1 + u) * sin(β)), u, (k * hx), ((k + 1) * hx), β, 0, α], k, 0, m - 1) → ShellsFront
  30. Sequence(Surface[-1 + ((1 + (k * hx)) * cos(β)), (v * g(((k + p) * hx))) + ((1 - v) * f(((k + p) * hx))), ((1 + (k * hx)) * sin(β)), v, 0, 1, β, 0, α], k, 0, m - 1) → ShellsSidesLeft
  31. Sequence(Surface[-1 + ((1 + ((k + 1) * hx)) * cos(β)), (v * g(((k + p) * hx))) + ((1 - v) * f(((k + p) * hx))), ((1 + ((k + 1) * hx)) * sin(β)), v, 0, 1, β, 0, α], k, 0, m - 1) → ShellsSidesRight