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Disc Method Demonstration

Disc Method Demonstration
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2017-04-20 02:00
FunctionSolid GeometryCalculusIntegralGeometric Construction
In collections volume of revolution
This example demonstrates the calculation of the volume of a solid of revolution using the disc method. It involves core mathematical concepts such as integration, volume calculation of rotated bodies, and function graphs. GeoGebra commands and objects like Sequence, Rotate, Intersect, Polygon, Circle, Surface, Cylinder, Dilate, Segment, Textfield, and Integral are used in the construction. Suitable for high school and above students to learn and understand the application of integration in volume calculations.

GeoGebra Commands & Parameters

  1. Sequence(Polygon[(a + (i * deltax), 0), (a + ((i + 1) * deltax), 0), (a + ((i + 1) * deltax), f(a + ((i + 1) * deltax))), (a + (i * deltax), f(a + ((i + 1) * deltax)))], i, 0, n - 1, 1) → list1
  2. Rotate(list1, α, y = 0) → list1'
  3. Intersect(i, l) → B
  4. Intersect(k, l) → C
  5. Intersect(j, k) → D
  6. Polygon(C, B, A, D) → poly1, c_2, b_1, a_1, d
  7. Rotate(C, α, y = 0) → C'
  8. Rotate(B, α, y = 0) → B'
  9. Rotate(A, α, y = 0) → A'
  10. Rotate(D, α, y = 0) → D'
  11. Polygon(C', B', A', D') → poly1', c', b', a', d'
  12. Sequence((a + (i * deltax), 0), i, 0, n, 1) → list2
  13. Sequence((a + (i * deltax), f(a + (i * deltax))), i, 1, n, 1) → list3
  14. Rotate(list3, α, y = 0) → list3'
  15. Sequence((a + (i * deltax), f(a + ((i + 1) * deltax))), i, 0, n - 1, 1) → list4
  16. Sequence(x = a + (i * deltax), i, 1, n, 1) → list6
  17. Sequence(Segment[Element[list3, i], Element[list4, i]], i, 0, n, 1) → list5
  18. Sequence(CircleSector[Element[list2, i + 1], Element[list3', i], Element[list3, i], xOyPlane], i, 1, n, 1) → list7
  19. Sequence(Cylinder[Element[list2, i], Element[list2, i + 1], f(Element[list2, i + 1])], i, 1, n, 1) → list8
  20. Sequence(Circle[Element[list2, i], f(Element[list2, i]), x = 0], i, 2, n + 1, 1) → list10
  21. Circle((a, 0), f(a + deltax), x = 0) → p
  22. Surface(q, α) → e
  23. Cylinder((x(A), 0), (x(A) - deltax, 0), y(A)) → o, r, s, t
  24. Dilate(E, If[0 < u ≤ 1, 1 - u, 0], D) → E'
  25. Dilate(F, If[0 < u ≤ 1, 1 - u, 0], A) → F'
  26. Segment(E, E') → f_1
  27. Segment(F, F') → g_1
  28. Dilate(D, If[1 < u ≤ 2, 2 - u, 0], A) → D'_1
  29. Segment(D, D'_1) → h_1
  30. Rotate(F, α, y = 0) → F'_1
  31. Rotate(F', α, y = 0) → F''
  32. Segment(F'_1, F'') → i_1
  33. Rotate(E, α, y = 0) → E'_1
  34. Rotate(E', α, y = 0) → E''
  35. Segment(E'_1, E'') → j_1
  36. Rotate(D, α, y = 0) → D'_2
  37. Rotate(D'_1, α, y = 0) → G
  38. Segment(D'_2, G) → k_1
  39. Textfield(f) → InputBox1
  40. Textfield(a) → InputBox2
  41. Textfield(b) → InputBox3
  42. Integral(f, a, b) → m_1

AI Drawing Prompt Reference

Create a 3D demonstration of the disc method for volume of revolution. Show a function curve on the x-axis from a to b, divided into n vertical slices. Each slice is rotated around the x-axis to form cylindrical discs or shells. Display the original curve, the rotated surface, several representative cylinders, circular cross-sections, and connecting segments between original and rotated points. Include an integral area indicator and input boxes for adjusting the function, interval, and number of slices. The scene illustrates how integration computes the volume of a solid of revolution.
The prompt above was compiled from the original drawing commands and is for reference only.