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Volume by Shells Example

Volume by Shells Example
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2016-03-18 09:10
Solid GeometryCalculusIntegral
In collections AP Calculus volume problems
This example demonstrates how to calculate the volume of a solid of revolution using the shell method. It involves core calculus concepts such as integral computation, volume of solids of revolution, and the application of the shell method. Key GeoGebra commands and objects used in the construction include Segment, Polygon, Rotate, Integral, Sequence, Surface, and RectangleSum to model the rotating solid, generate shells, and compute the integral. Suitable for high school and above students learning calculus, particularly those needing to understand methods for calculating volumes of revolution.

GeoGebra Commands & Parameters

  1. Segment((0, 0, 0), (0, 6, 0)) → k
  2. Polygon((0, (DK * h), 0), (0, ((DK + 1) * h), 0), (H(((DK + p) * h)), ((DK + 1) * h), 0), (H(((DK + p) * h)), (DK * h), 0)) → Rectangle1, b, c, d, e
  3. Rotate(Rectangle1, α, a) → Rectangle1'
  4. Integral((((2 * π) * R(y)) * H(y)), 0, 6) → VolInteg
  5. Sequence(Polygon[(0, (k * h), 0), (0, ((k + 1) * h), 0), (H(((k + p) * h)), ((k + 1) * h), 0), (H(((k + p) * h)), (k * h), 0)], k, 0, m - 1) → Rectangles
  6. Rotate(Rectangles, α, a) → Rectangles'
  7. Rotate(g_1, α, a) → i
  8. Rotate(f_1, α, a) → j
  9. Surface(u, 7 + ((7 - g(u)) * cos(β)), ((7 - g(u)) * sin(β)), u, 0, 9, β, 0, (2 * π)) → Inner
  10. Surface(u, 7 - ((7 - f(u)) * cos(β)), ((7 - f(u)) * sin(β)), u, 0, 9, β, 0, (2 * π)) → Outer
  11. Rotate(k, α, a) → k'
  12. RectangleSum((((2 * π) * R(x)) * H(x)), 0, (m * h), m, p) → VSh
  13. Sequence(Surface[(u * H(((k + p) * h))), 7 - ((7 - (k * h)) * cos(β)), ((7 - (k * h)) * sin(β)), u, 0, 1, β, 0, α], k, 0, m - 1) → Outers
  14. Sequence(Surface[(u * H(((k + p) * h))), 7 - ((7 - ((k + 1) * h)) * cos(β)), ((7 - ((k + 1) * h)) * sin(β)), u, 0, 1, β, 0, α], k, 0, m - 1) → Inners
  15. Sequence(Surface[0, 7 - ((7 - ((k + u) * h)) * cos(β)), ((7 - ((k + u) * h)) * sin(β)), u, 0, 1, β, 0, α], k, 0, m - 1) → SidesLeft
  16. Sequence(Surface[H(((k + p) * h)), 7 - ((7 - ((k + u) * h)) * cos(β)), ((7 - ((k + u) * h)) * sin(β)), u, 0, 1, β, 0, α], k, 0, m - 1) → SidesRight
  17. Surface((u * H(((DK + p) * h))), 7 - ((7 - ((DK + 1) * h)) * cos(β)), ((7 - ((DK + 1) * h)) * sin(β)), u, 0, 1, β, 0, α) → Inner1
  18. Surface((u * H(((DK + p) * h))), 7 - ((7 - (DK * h)) * cos(β)), ((7 - (DK * h)) * sin(β)), u, 0, 1, β, 0, α) → Outer1
  19. Surface(0, 7 - ((7 - ((DK + u) * h)) * cos(β)), ((7 - ((DK + u) * h)) * sin(β)), u, 0, 1, β, 0, α) → SideLeft1
  20. Surface(H(((DK + p) * h)), 7 - ((7 - ((DK + u) * h)) * cos(β)), ((7 - ((DK + u) * h)) * sin(β)), u, 0, 1, β, 0, α) → SideRight1