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Disc Action!!! Volumes of Revolution: The Disc Method

Disc Action!!! Volumes of Revolution: The Disc Method
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2017-01-17 22:04
FunctionSolid GeometryCalculusIntegralSlider Animation
In collections BC Calculus GeoGebra 3D & AR: PreCalc & Calculus Resources Calculus Concepts Volume of Revolution
① This dynamic and modifiable GeoGebra applet vividly demonstrates the disc method for computing volumes of solids of revolution, where a plane region is rotated about an axis to form a 3D solid. ② The core mathematical concept is the application of definite integrals to calculate volume, specifically by slicing the solid into thin circular discs and summing their volumes via integration, involving functions, intervals, axis of rotation, and volume elements. ③ Construction highlights: from the construction summary, commands such as Sequence, Rotate, Intersect, Polygon, Circle, Surface, Cylinder, Dilate, Segment, Textfield, and Integral are used to build the function f, interval [a,b], number of subdivisions n, thickness deltax, a list of discs (list1), and the resulting surface, with sliders and text fields for interactive parameter adjustment. ④ Suitable for advanced high school to early college calculus students to grasp the geometric meaning of the volume formula, and for teachers as a classroom demonstration or for self-directed exploration.

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, xAxis) → 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 diagram illustrating the disc method for volumes of revolution. Show a function curve over an interval [a,b] divided into n equal subintervals of width Δx. Draw vertical rectangles from the x-axis to the curve at each subdivision. Rotate these rectangles around the x-axis to form a series of cylindrical discs, and display the resulting surface of revolution. Use semi-transparent discs or cylinders to represent each slice's volume, and indicate the integral area. Include input fields for the function, interval endpoints, and number of subdivisions, plus a slider for rotation angle.
The prompt above was compiled from the original drawing commands and is for reference only.