MathPlotAI Logo MathPlotAI

Rotational Cross Section Volume

Rotational Cross Section Volume
Loading the math board and drawing, please wait…
2016-11-11 05:09
Solid GeometryCalculusIntegral
In collections Volume Apps
This example demonstrates how to calculate volume by rotating cross sections. Key mathematical concepts include integration and solid geometry. Using GeoGebra commands like Simplify, Integral, and Surface, the rotation of the solid can be constructed. Suitable for high school or university calculus and solid geometry courses.

GeoGebra Commands & Parameters

  1. Simplify(ISOSarea) → h1
  2. Integral(isosarea, x) → i
  3. Simplify(SQarea) → p1
  4. Integral(area2dfunction, x) → j
  5. Textfield(f) → Top
  6. Textfield(xlower) → LB
  7. Segment(C, B) → cbsq
  8. Segment(B, D) → dbsq
  9. Segment(E, C) → ecsq
  10. Integral(p1, x) → p2
  11. Simplify(p3(y) - p4(y)) → p5
  12. Simplify(p3) → f_2
  13. Textfield(g) → Bottom
  14. Textfield(xupper) → UB
  15. Intersect(f, g, xlower, xupper) → F, G
  16. Surface(t, (u * f(t)) + ((1 - u) * g(t)), 0, t, xlower, xupper, u, 0, 1) → k
  17. Segment(B, Disos) → disos
  18. Segment(C, Disos) → hyp
  19. Segment(B, C) → cisos
  20. Segment(C, B) → ceq
  21. Segment(B, H) → eq1
  22. Segment(C, H) → eq2
  23. Integral(h1, x) → h2
  24. Simplify(h3(y) - h4(y)) → h5
  25. Simplify(EQarea) → e1
  26. Integral(e1, x) → e2
  27. Integral(eqarea, x) → e
  28. Segment(E, D) → edsq
  29. Simplify(e3(y) - e4(y)) → e5
  30. Semicircle(B, C, xOyPlane) → c
  31. Rotate(c, 1.5707963267948966, Segment[B, C]) → semi
  32. Simplify(SCarea) → sc1
  33. Integral(scarea, x) → s
  34. Integral(sc1, x) → sc2
  35. Simplify(sc3(y) - sc4(y)) → sc5
  36. Segment(C, B) → cbsc
  37. Surface(u, (g(u) * v) + ((f(u) + g(u)) / 2 * (1 - v)), (((1 - v) * (f(u) - g(u))) * sqrt(3) / 2), u, xlower, xval, v, 0, 1) → a
  38. Surface(u, (f(u) * v) + ((f(u) + g(u)) / 2 * (1 - v)), (((1 - v) * (f(u) - g(u))) * sqrt(3) / 2), u, xlower, xval, v, 0, 1) → b
  39. Surface(u, (v * f(u)) + ((1 - v) * g(u)), (abs(f(u) - g(u)) / 2 * sqrt(1 - ((2 * v) - 1)^(2))), u, xlower, xval, v, 0, 1) → d
  40. Surface(u, g(u), (v * abs(f(u) - g(u))), u, xlower, xval, v, 0, 1) → h
  41. Surface(u, (v * g(u)) + ((1 - v) * f(u)), (v * abs(f(u) - g(u))), u, xlower, xval, v, 0, 1) → l
  42. Surface(u, g(u), (v * abs(f(u) - g(u))), u, xlower, xval, v, 0, 1) → m
  43. Surface(u, f(u), (v * abs(f(u) - g(u))), u, xlower, xval, v, 0, 1) → n
  44. Surface(u, (v * g(u)) + ((1 - v) * f(u)), abs(f(u) - g(u)), u, xlower, xval, v, 0, 1) → o
  45. Length(cbsq) → baselength
  46. Segment(I, J) → divider1
  47. Segment(K, L) → divider2
  48. Textfield(xinc) → inputBox1

AI Drawing Prompt Reference

Draw a solid of revolution by rotating cross sections of plane figures. Start by drawing two parallel rectangles and rotating one around its side to form a cylinder. Then, draw two isosceles triangles and rotate them around their bases to form two cones. Finally, combine these shapes to form a complex solid and calculate its volume.
The prompt above was compiled from the original drawing commands and is for reference only.