MathPlotAI Logo MathPlotAI

Volumes - Discs (Perpendicular to the x-axis)

Volumes - Discs (Perpendicular to the x-axis)
Loading the math board and drawing, please wait…
2020-04-09 11:04
Solid GeometryCalculusIntegral
In collections Calculus - Applications of Integrals - Areas and Volumes
This example demonstrates the calculation of volumes of solids of revolution using the disc method. Core mathematical concepts include integral calculus, volume formulas for rotated bodies, and geometric transformations of function graphs. The construction utilizes GeoGebra commands and objects such as Sequence, Rotate, Intersect, Polygon, Circle, Surface, Cylinder, Dilate, Segment, Textfield, and Integral. It is suitable for high school and above students to understand the method of calculating volumes of solids of revolution and to visualize dynamic effects.

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

Draw a demonstration of calculating the volume of a solid of revolution using the disc method. The image should feature a function graph rotated around the x-axis, represented by a series of stacked discs. Each disc has a base on the x-axis and a height equal to the function's value, with the integral bounds and calculation result labeled to illustrate the volume integration process.
The prompt above was compiled from the original drawing commands and is for reference only.