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Volume by Cylindrical Shells - Example: Torus

Volume by Cylindrical Shells - Example: Torus
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2016-03-15 11:07
FunctionSolid GeometryCalculusIntegralGeometric Construction
This example demonstrates the calculation of the volume of a torus using the method of cylindrical shells. Core mathematical concepts include integration, volume calculation, and the cylindrical shells method. The construction utilizes GeoGebra commands and objects such as Sequence, Surface, Polygon, and Segment to dynamically visualize the generation of the torus, providing an intuitive understanding of the geometric interpretation of integration. Suitable for high school and above students learning about volume calculation methods in calculus.

GeoGebra Commands & Parameters

  1. Sequence(a + ((k - 0.5) * dx), k, 1, n) → midpts
  2. Sequence(a + (k * dx), k, 0, n - 1) → leftpts
  3. Sequence(a + (k * dx), k, 1, n) → rightpts
  4. Sequence(Polygon[(xk(k - 1), y_B(Element[evalpts, k])), (xk(k - 1), y_T(Element[evalpts, k])), (xk(k), y_T(Element[evalpts, k])), (xk(k), y_B(Element[evalpts, k]))], k, 1, n) → Rectangles
  5. Sequence((Element[evalpts, k], 0), k, 1, n) → evalptsplot
  6. Surface(xrot + ((xk(k - 1) - xrot) * cos(t)), h, ((xk(k - 1) - xrot) * sin(t)), t, 0, (2 * π), h, y_B(Element[evalpts, k]), y_T(Element[evalpts, k])) → ShellIn_k
  7. Surface(xrot + (r * cos(t)), y_T(Element[evalpts, k]), (r * sin(t)), t, 0, (2 * π), r, xk(k - 1) - xrot, xk(k) - xrot) → UpShell_k
  8. Surface(xrot + ((xk(k) - xrot) * cos(t)), h, ((xk(k) - xrot) * sin(t)), t, 0, (2 * π), h, y_B(Element[evalpts, k]), y_T(Element[evalpts, k])) → ShellOut_k
  9. Surface(xrot + (r * cos(t)), y_B(Element[evalpts, k]), (r * sin(t)), t, 0, (2 * π), r, xk(k - 1) - xrot, xk(k) - xrot) → ShellDown_k
  10. Sequence(Surface[xrot + ((xk(k - 1) - xrot) * cos(t)), h, ((xk(k - 1) - xrot) * sin(t)), t, 0, (2 * π), h, y_B(Element[evalpts, k]), y_T(Element[evalpts, k])], k, 1, n) → ShellsIn
  11. Sequence(Surface[xrot + (r * cos(t)), y_T(Element[evalpts, k]), (r * sin(t)), t, 0, (2 * π), r, xk(k - 1) - xrot, xk(k) - xrot], k, 1, n) → ShellsUp
  12. Sequence(Surface[xrot + (r * cos(t)), y_B(Element[evalpts, k]), (r * sin(t)), t, 0, (2 * π), r, xk(k - 1) - xrot, xk(k) - xrot], k, 1, n) → ShellsDown
  13. Sequence(Surface[xrot + ((xk(k) - xrot) * cos(t)), h, ((xk(k) - xrot) * sin(t)), t, 0, (2 * π), h, y_B(Element[evalpts, k]), y_T(Element[evalpts, k])], k, 1, n) → ShellsOut
  14. Surface(xrot + (cos((-t)) * (u - xrot)), y_B(u), (sin((-t)) * (u - xrot)), u, a, b, t, 0, RotationAngle) → Solid_B
  15. Surface(xrot + (cos((-t)) * (u - xrot)), y_T(u), (sin((-t)) * (u - xrot)), u, a, b, t, 0, RotationAngle) → Solid_T
  16. Surface(xrot + ((a - xrot) * cos((-t))), h, ((a - xrot) * sin((-t))), t, 0, RotationAngle, h, y_B(a), y_T(a)) → Solid_a
  17. Surface(xrot + ((b - xrot) * cos((-t))), h, ((b - xrot) * sin((-t))), t, 0, RotationAngle, h, y_B(b), y_T(b)) → Solid_b
  18. Surface(xrot + (cos((-RotationAngle)) * (u - xrot)), ((1 - w) * y_B(u)) + (w * y_T(u)), (sin((-RotationAngle)) * (u - xrot)), w, 0, 1, u, a, b) → Solid_e
  19. Surface(xrot + u - xrot, ((1 - w) * y_B(u)) + (w * y_T(u)), (0 * (u - xrot)), w, 0, 1, u, a, b) → region3d
  20. Polygon((xk(k - 1), y_B(Element[evalpts, k])), (xk(k - 1), y_T(Element[evalpts, k])), (xk(k), y_T(Element[evalpts, k])), (xk(k), y_B(Element[evalpts, k]))) → Rectangle_k, f, g, h, i
  21. Segment((0, 0), (xrot, 0)) → c