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AB2011-3b Volume Problem

AB2011-3b Volume Problem
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2016-03-18 02:41
CalculusIntegralGeometric Construction
This is an AP Calculus AB2011-3b problem demonstrating how to calculate the volume of a solid of revolution. The example involves the concept of integration in calculus, showing the application of integration in geometry. It uses commands like Sequence, Intersect, Segment, Locus, Integral, etc., suitable for high school calculus courses.

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

AI Drawing Prompt Reference

Draw a solid of revolution formed by rotating the area enclosed by the curves y=x^2 and y=6-x around the x-axis. First, draw the two curves, then find their intersection point, and draw the cross-sections of the solid. Use integration to calculate the volume and display the surface of the solid.
The prompt above was compiled from the original drawing commands and is for reference only.