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Volume with Known Cross Section

Volume with Known Cross Section
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2019-03-15 11:40
Solid GeometryCalculusIntegralTransformationGeometric Construction
This example demonstrates how to calculate the volume of a solid with a known cross-sectional area. It involves core mathematical concepts such as volume calculation, similarity transformation, similarity, dilation, incircle, tree diagrams, and area. The construction utilizes GeoGebra commands like SelectedElement, IndexOf, Intersect, Segment, Point, CurveCartesian, Polygon, Surface, Midpoint, OrthogonalPlane, PointIn, Line, Circle, IntegralBetween, and Rotate. It is suitable for high school and above students to understand the practical application of volume calculation and geometric transformations.

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