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AB2009_4b_B: 2009 AP Test B, Problem 4

AB2009_4b_B: 2009 AP Test B, Problem 4
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2016-03-19 11:23
Solid GeometryCalculusIntegral
This example demonstrates the volume calculation of a solid with a square cross-section, as part of the 2009 AP Test B, Problem 4. Using GeoGebra commands such as Intersect, Segment, and Integral, the example dynamically shows the solid. Suitable for high school mathematics education, particularly for calculus and solid geometry.

GeoGebra Commands & Parameters

  1. Intersect(f, g, 3, 5) → E
  2. Segment(A, B) → e
  3. Integral((g(x) - f(x))^(2), 0, 4) → Integ
  4. Locus(C, s) → loc1
  5. If(togS ≟ 1, "Hide surface", "Show surface") → ShowSurface
  6. Locus(D, s) → loc2
  7. Polygon(A, B, C, D) → Rect, a, b, c, d
  8. Angle(B, A, D) → α
  9. Segment((0, 6, 27), (9, 6, 27)) → i
  10. Surface(u, (v * f(u)) + ((1 - v) * g(u)), 0, u, 0, 4, v, 0, 1) → Base
  11. Surface(u, (v * f(u)) + ((1 - v) * g(u)), g(u) - f(u), u, 0, 4, v, 0, 1) → Top
  12. Surface(u, f(u), (v * (g(u) - f(u))), u, 0, 4, v, 0, 1) → SurfaceRight
  13. Surface(u, g(u), (v * (g(u) - f(u))), u, 0, 4, v, 0, 1) → SurfaceLeft
  14. If(togDF ≟ 1, "Show Directions", "Show volume integral") → ToggleDirectionVolumeIntegral
  15. Sequence(Polygon[((k * h), f(((k + p) * h)), 0), ((k * h), g(((k + p) * h)), 0), ((k * h), g(((k + p) * h)), ht(((k + p) * h))), ((k * h), f(((k + p) * h)), ht(((k + p) * h)))], k, 0, m - 1) → SlicesLeft
  16. Sequence(Polygon[(((k + 1) * h), f(((k + p) * h)), 0), (((k + 1) * h), g(((k + p) * h)), 0), (((k + 1) * h), g(((k + p) * h)), ht(((k + p) * h))), (((k + 1) * h), f(((k + p) * h)), ht(((k + p) * h)))], k, 0, m - 1) → SlicesRight
  17. Sequence(Surface[((k + u) * h), (v * f(((k + p) * h))) + ((1 - v) * g(((k + p) * h))), ht(((k + p) * h)), u, 0, 1, v, 0, 1], k, 0, m - 1) → SlicesTop
  18. Sequence(Surface[((k + u) * h), f(((k + p) * h)), (v * ht(((k + p) * h))), u, 0, 1, v, 0, 1], k, 0, m - 1) → SlicesFront
  19. Sequence(Surface[((k + u) * h), g(((k + p) * h)), (v * ht(((k + p) * h))), u, 0, 1, v, 0, 1], k, 0, m - 1) → SlicesBack
  20. Sequence(Polygon[((k * h), f(((k + p) * h)), 0), ((k * h), g(((k + p) * h)), 0), (((k + 1) * h), g(((k + p) * h)), 0), (((k + 1) * h), f(((k + p) * h)), 0)], k, 0, m - 1) → SlicesBase
  21. Sequence(Area[Element[SlicesLeft, k]], k, 1, m) → Areas