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Volumes by Cylindrical Shells

Volumes by Cylindrical Shells
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2016-03-14 02:28
CalculusIntegralGeometric Construction
This example demonstrates how to calculate the volume of a solid of revolution using the method of cylindrical shells. Key concepts include integration, volume of solids of revolution, and the method of cylindrical shells. It constructs cylindrical shells, rotation axis, and regions to illustrate the geometric meaning of the method. Suitable for calculus courses.

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

AI Drawing Prompt Reference

Draw a diagram illustrating the volume of a solid of revolution using the method of cylindrical shells. Start by drawing the rotation axis xrot, then draw a series of rectangles representing the bases of the cylindrical shells. Next, draw the cylindrical shells formed by rotating these rectangles around the axis, including both inner and outer shells. Finally, draw the boundaries of the solid, including the upper and lower boundaries.
The prompt above was compiled from the original drawing commands and is for reference only.