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Volume of Revolution by Cylindrical Shells

Volume of Revolution by Cylindrical Shells
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2017-01-15 09:54
CalculusIntegralGeometric Construction
In collections BC Calculus Calculus - Applications of Integrals - Areas and Volumes
This example demonstrates the volume calculation of a solid of revolution using the method of cylindrical shells. Key mathematical concepts include integration and calculus. It utilizes commands such as polynomial, integral, sequence, and polygon in GeoGebra. Suitable for high school and above students to learn the method of calculating the volume of a solid of revolution.

GeoGebra Commands & Parameters

  1. Point(x = 0) → A
  2. Point(x = 2) → B
  3. Point(x = 4) → C
  4. Polynomial({A, B, C}) → f
  5. Integral(h, x(A), x(C)) → a
  6. Point(h) → D
  7. Sequence(Polygon[((i * deltax), 0), (((i + 1) * deltax), 0), (((i + 1) * deltax), h(((i + 1) * deltax))), ((i * deltax), h(((i + 1) * deltax)))], i, 0, n - 1, 1) → list1
  8. Polygon(D, (x(D) - deltax, y(D)), (x(D) - deltax, 0), (x(D), 0)) → poly1, d, i, j, k
  9. Line(E, F) → l
  10. Rotate(h, α, l) → h_1
  11. Surface((t * cos(u)), h(x(D)), (t * sin(u)), t, x(D) - deltax, x(D), u, 0, α) → e
  12. Vector(D, (x(D), 0, 0)) → u
  13. Surface((t * cos(u)), 0, (t * sin(u)), t, x(D) - deltax, x(D), u, 0, α) → m
  14. Cylinder((0, 0, 0), (0, y(D), 0), x(D)) → o, p, q, r
  15. Cylinder((0, 0, 0), (0, y(D), 0), x(D) - deltax) → s, t, c_1, v
  16. Surface((t * cos(u)), h(t), (t * sin(u)), t, 0, 10, u, 0, α) → b
  17. Rotate(list1, (-α), l) → list1'
  18. Rotate(poly1, (-α), l) → poly1'
  19. Dilate(G, If[0 < w ≤ 1, 1 - w, 0], D) → G'
  20. Segment(G, G') → f_1
  21. Rotate(D, π, l) → D'
  22. Rotate((x(D) - deltax, y(D)), π, l) → J
  23. Dilate(H, If[1 < w ≤ 2, 2 - w, 0], I) → H'
  24. Segment(H, H') → g_1
  25. Dilate(D', If[2 < w ≤ 3, 3 - w, 0], J) → J'
  26. Segment(D', J') → i_1