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Washer Volume Demonstration

Washer Volume Demonstration
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2017-01-19 00:48
Solid GeometryCalculusIntegral
This example demonstrates the volume of a washer using dynamic and modifiable illustrations, involving key concepts such as washer volume and integration. It utilizes commands like Dilate, Cylinder, and Circle in GeoGebra. Suitable for high school and above students studying solid geometry and calculus.

GeoGebra Commands & Parameters

  1. Dilate(B, If[0 < j ≤ 1, 1 - j, 0], A) → B'
  2. Segment(B, B') → l
  3. Cylinder(B, B', r) → a, c, d, b
  4. Dilate(E, If[2 < j ≤ 3, 3 - j, 0], C) → E'
  5. Segment(E, E') → f
  6. Dilate(G, If[1 < j ≤ 2, 2 - j, 0], C) → G'
  7. Dilate(F, If[1 < j ≤ 2, 2 - j, 0], E) → F'
  8. Segment(G, G') → g
  9. Segment(F, F') → i
  10. Dilate(A, If[3 < j ≤ 4, 4 - j, 0], H) → A'
  11. Dilate(A, If[3 < j ≤ 4, 4 - j, 0], D) → A'_1
  12. Circle(A', A'_1, I) → k
  13. Dilate(A, If[4 < j ≤ 5, 5 - j, 0], B) → A'_2
  14. Cylinder(A, A'_2, e) → m, p, q, n
  15. Dilate(G, If[1 < j ≤ 2, 2 - j, 0], A) → A'_3
  16. Segment(G, A'_3) → s
  17. Dilate(A, If[6 < j ≤ 7, 7 - j, 0], B) → A'_4
  18. Vector(A, A'_4) → u
  19. Translate(m, u) → m'
  20. Cylinder(A, A'_4, e) → o, t, c_1, v
  21. Dilate(B, If[7 < j ≤ 8, 8 - j, 0], J) → B'_1
  22. Vector(B, B'_1) → w
  23. Translate(m', w) → m''
  24. Dilate(J, If[8 < j ≤ 9, 9 - j, 0], K) → J'
  25. Segment(J, J') → f_1
  26. Dilate(C, If[9 < j ≤ 10, 10 - j, 0], L) → C'
  27. Vector(C, C') → u_1
  28. Translate(E, u_1) → E'_1
  29. Translate(E', u_1) → E''
  30. Segment(E'_1, E'') → f'
  31. Translate(G, u_1) → G'_1
  32. Translate(G', u_1) → G''
  33. Segment(G'_1, G'') → g'
  34. Translate(F, u_1) → F'_1
  35. Translate(F', u_1) → F''
  36. Segment(F'_1, F'') → i'
  37. Dilate(K, If[9 < j ≤ 10, 10 - j, 0], M) → K'
  38. Dilate(O, If[9 < j ≤ 10, 10 - j, 0], N) → O'
  39. Segment(K, K') → g_1
  40. Segment(O, O') → h_1
  41. Rotate(J, π, Line[(0, 0, 0), (0, 0, 1)]) → J'_1
  42. Rotate(J', π, Line[(0, 0, 0), (0, 0, 1)]) → J''
  43. Segment(J'_1, J'') → i_1
  44. Rotate(G, π, Line[(0, 0, 0), (0, 0, 1)]) → G'_2
  45. Rotate(A'_3, π, Line[(0, 0, 0), (0, 0, 1)]) → P
  46. Segment(G'_2, P) → s'

AI Drawing Prompt Reference

Create a 3D dynamic illustration of a washer volume. Show multiple cylinders, some truncated or translated, forming washer-like shapes. Include a central circular cross-section and several cylinders arranged in different directions, with some translated copies. Add line segments connecting key points to indicate radii or heights. The overall scene demonstrates the integration process for volume calculation, with adjustable colors and transparency for clarity.
The prompt above was compiled from the original drawing commands and is for reference only.