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2009 AB Calculus Problem 4c: Square Cross Sections Volume

2009 AB Calculus Problem 4c: Square Cross Sections Volume
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2016-03-18 22:39
Solid GeometryCalculusIntegral
In collections AP Calculus volume problems
This example demonstrates problem 4c from the 2009 AP AB Calculus exam, showing a solid with square cross-sections. Key mathematical concepts include integration and solid geometry. Using commands like Intersect, Segment, Integral, Locus, etc., the volume calculation process is dynamically illustrated. Suitable for high school calculus education and learning.

GeoGebra Commands & Parameters

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

AI Drawing Prompt Reference

Draw a region bounded by the curves y = sqrt(x) and y = x/2, with the top being a rectangular plane parallel to the x-axis and height 4. Then, build a series of square cross-sections perpendicular to the x-axis using these rectangles as bases to form a solid. Finally, calculate the volume of this solid.
The prompt above was compiled from the original drawing commands and is for reference only.