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Volumes of Solids with Different Cross Sections

Volumes of Solids with Different Cross Sections
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2019-04-12 07:51
TrigonometryCoordinate GeometrySolid GeometryCalculusGeometric Construction
This example demonstrates how to visualize and calculate the volume of solids with varying cross sections. It involves core math concepts like volume calculation, right triangles, trigonometry, similar triangles, and solid geometry. GeoGebra commands such as Segment, Point, Polygon, and IntegralBetween are used in the construction. Suitable for high school students and math enthusiasts exploring 3D geometry volume methods.

GeoGebra Commands & Parameters

  1. SelectedElement(C) → text1
  2. IndexOf(text1, C) → a
  3. Intersect(f, g, 1) → A
  4. Intersect(f, g, 2) → B
  5. Segment((x(A), -3), (x(B), -3)) → h
  6. Point(h) → D
  7. Segment(E, F) → i
  8. CurveCartesian(t, g(t), p(t), t, x(A), x(B)) → b
  9. CurveCartesian(t, f(t), p(t), t, x(A), x(B)) → c
  10. Polygon(E, F, (x(F), f(x(F)), p(x(F))), (x(F), g(x(F)), p(x(F)))) → q1, e, f_1, j, k
  11. Surface(t, g(t), (u * p(t)), t, x(A), x(B), u, 0, 1) → d
  12. Surface(t, f(t), (u * p(t)), t, x(A), x(B), u, 0, 1) → l
  13. Surface(t, (u * g(t)) + ((1 - u) * f(t)), p(t), t, x(A), x(B), u, 0, 1) → m
  14. Polygon(E, F, F + (0, 0, e), E + (0, 0, e)) → q2, e_1, f_2, n, q
  15. CurveCartesian(t, g(t), f(t) - g(t), t, x(A), x(B)) → o
  16. CurveCartesian(t, f(t), f(t) - g(t), t, x(A), x(B)) → r
  17. Surface(t, g(t), (u * (f(t) - g(t))), t, x(A), x(B), u, 0, 1) → s
  18. Surface(t, f(t), (u * (f(t) - g(t))), t, x(A), x(B), u, 0, 1) → t
  19. Surface(t, (u * f(t)) + ((1 - u) * g(t)), f(t) - g(t), t, x(A), x(B), u, 0, 1) → u
  20. Midpoint(i) → G
  21. Surface(t, ((f(t) - g(t)) / 2 * cos(u)) + (f(t) + g(t)) / 2, ((f(t) - g(t)) / 2 * sin(u)), t, x(A), x(B), u, 0, (2 * π)) → v
  22. OrthogonalPlane(G, i) → w
  23. PointIn(w) → H
  24. Line(G, H) → g_1
  25. Circle(G, (f(x(G)) - g(x(G))) / 2, g_1) → c_1
  26. IntegralBetween(f, g, x(A), x(B)) → a_1
  27. CurveCartesian(t, (f(t) + g(t)) / 2, ((f(t) - g(t)) / 2 * sqrt(3)), t, x(A), x(B)) → bb
  28. Surface(t, (u * g(t)) + ((1 - u) * y(bb(t))), ((1 - u) * z(bb(t))), t, x(A), x(B), u, 0, 1) → b_1
  29. Surface(t, (u * f(t)) + ((1 - u) * y(bb(t))), ((1 - u) * z(bb(t))), t, x(A), x(B), u, 0, 1) → d_1
  30. Rotate(Polygon[E, F, 3], π / 2, e) → t1
  31. CurveCartesian(t, f(t), f(t) - g(t), t, x(A), x(B)) → hh
  32. Polygon(E, F, F + (0, 0, f(x(F)) - g(x(F)))) → t2, i_1, e_2, f_3
  33. Surface(t, (u * g(t)) + ((1 - u) * f(t)), ((1 - u) * z(hh(t))), t, x(A), x(B), u, 0, 1) → h_1
  34. Surface(t, f(t), (u * z(hh(t))), t, x(A), x(B), u, 0, 1) → j_1
  35. Surface(t, ((f(t) - g(t)) / 2 * cos(u)) + (f(t) + g(t)) / 2, ((f(t) - g(t)) / 2 * sin(u)), t, x(A), x(B), u, 0, π) → k_1
  36. Rotate(Semicircle[E, F], π / 2, e) → p_1

AI Drawing Prompt Reference

Draw a solid with varying cross-sections. First, plot two curves f and g. Then, create a series of rectangular cross-sections parallel to the x-axis between these curves. Use these cross-sections to construct the solid surface. Finally, calculate and display the volume enclosed by the two curves.
The prompt above was compiled from the original drawing commands and is for reference only.