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2013 AP Test Problem 5c: Volume of a Square Prism

2013 AP Test Problem 5c: Volume of a Square Prism
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2016-03-19 11:43
Solid GeometryCalculusIntegral
In collections AP Calculus volume problems
This example illustrates the 2013 AP Test Problem 5c, focusing on the calculation of the volume of a square prism in solid geometry. Using GeoGebra commands such as Polygon, Segment, and Integral, the construction process and volume changes of the square prism are dynamically demonstrated. Suitable for high school mathematics education, particularly in calculus and solid geometry courses.

GeoGebra Commands & Parameters

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