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Solids with Different Cross Sections

Solids with Different Cross Sections
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2018-02-16 03:33
FunctionCoordinate GeometrySolid GeometryCalculusIntegral
This example demonstrates the construction and variation of solids with different cross-sectional shapes, involving core mathematical concepts such as solid geometry, volume calculation, continuity, integration, logarithmic functions, and geometric mean. It utilizes GeoGebra commands like SelectedElement, IndexOf, Intersect, Segment, Point, CurveCartesian, Polygon, Surface, Midpoint, OrthogonalPlane, PointIn, Line, Circle, IntegralBetween, Rotate. Suitable for high school and above students and teachers for teaching demonstrations and concept understanding.

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