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2011 AP Calculus AB Form B Problem 3c: Volume Problem

2011 AP Calculus AB Form B Problem 3c: Volume Problem
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2016-06-26 22:24
CalculusIntegralParametric EquationGeometric ConstructionSlider Animation
In collections Volumes in AP Calculus Volumes in AP Calculus
This example demonstrates the volume problem from the 2011 AP Calculus AB Form B Problem 3c. The core mathematical concepts include integration in calculus. Using GeoGebra commands such as Sequence, Intersect, Segment, Locus, Integral, etc., a geometric figure involving similar triangles and parametric curves is constructed. Suitable for high school calculus courses and learning.

GeoGebra Commands & Parameters

  1. Sequence(Polygon[(fI(((k + p) * h)), (k * h), 0), (g(((k + p) * h)), (k * h), 0), (g(((k + p) * h)), (k * h), ((2 * (k + p)) * h)), (fI(((k + p) * h)), (k * h), ((2 * (k + p)) * h))], k, 0, m - 1) → SlicesFront
  2. Intersect(f, g, 3, 5) → E
  3. Segment(A, B) → e
  4. Locus(C, s) → loc1
  5. Integral(((2 * y) * (6 - y - y^(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. Polygon(A, B, C, D) → Rect, a, b, c, d
  10. Angle(B, A, D) → α
  11. Surface(u^(2), u, (v * (2 * u)), u, 0, 2, v, 0, 1) → SurfaceLeft
  12. Surface(6 - u, u, (v * (2 * u)), u, 0, 2, v, 0, 1) → SurfaceRight
  13. Surface((v * u^(2)) + ((1 - v) * (6 - u)), u, 0, u, 0, 2, v, 0, 1) → Base
  14. Surface((v * u^(2)) + ((1 - v) * (6 - u)), u, (2 * u), u, 0, 2, v, 0, 1) → SurfaceFront
  15. Sequence(Surface[fI(((k + p) * h)), (k * h) + (v * h), ((u * (k + p)) * h), v, 0, 1, u, 0, 2], k, 0, m - 1) → SlicesLeft
  16. Sequence(Surface[u, v, ((2 * (k + p)) * h), v, (k * h), ((k + 1) * h), u, fI(((k + p) * h)), g(((k + p) * h))], k, 0, m - 1) → SlicesTop
  17. Sequence(Surface[g(((k + p) * h)), (k * h) + (v * h), ((u * (k + p)) * h), v, 0, 1, u, 0, 2], k, 0, m - 1) → SlicesRight
  18. Sequence(Area[Element[SlicesFront, k]], k, 1, m) → Areas
  19. Sequence(Polygon[(fI(((k + p) * h)), (k * h), 0), (g(((k + p) * h)), (k * h), 0), (g(((k + p) * h)), ((k + 1) * h), 0), (fI(((k + p) * h)), ((k + 1) * h), 0)], k, 0, m - 1) → SlicesBase
  20. Sequence(Polygon[(fI(((k + p) * h)), ((k + 1) * h), 0), (g(((k + p) * h)), ((k + 1) * h), 0), (g(((k + p) * h)), ((k + 1) * h), ((2 * (k + p)) * h)), (fI(((k + p) * h)), ((k + 1) * h), ((2 * (k + p)) * h))], k, 0, m - 1) → SlicesBack