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AB2008_1c_B Volume, Square Cross Sections

AB2008_1c_B Volume, Square Cross Sections
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2016-03-18 22:28
Solid GeometryCalculusIntegralGeometric ConstructionSlider Animation
This example demonstrates the volume of a solid formed by rotating a region, using square cross sections. It involves core calculus concepts like volume calculation, integration, limits, and geometric construction. GeoGebra commands used include Sequence, Intersect, Segment, Integral, Locus, Polygon, etc. Suitable for high school calculus students to understand volume calculation of solids of revolution.

GeoGebra Commands & Parameters

  1. 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
  2. Intersect(f, g, 3, 5) → E
  3. Segment(A, B) → e
  4. Integral((fI(x) - gI(x))^(2), 0, 3) → Integ
  5. Locus(C, s) → loc1
  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. If(togDF ≟ 1, "Show Directions", "Show volume integral") → ToggleDirectionVolumeIntegral
  12. Sequence(Area[Element[SlicesFront, k]], k, 1, m) → Areas
  13. 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
  14. Sequence(Surface[(v * fI(((k + p) * h))) + ((1 - v) * gI(((k + p) * h))), ((k + u) * h), ht(((k + p) * h)), u, 0, 1, v, 0, 1], k, 0, m - 1) → SlicesTop
  15. 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
  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[fI(((k + p) * h)), ((k + u) * h), (v * ht(((k + p) * h))), u, 0, 1, v, 0, 1], k, 0, m - 1) → SlicesRight
  18. Surface((v * fI(u)) + ((1 - v) * gI(u)), u, 0, u, 0, 3, v, 0, 1) → Base
  19. Surface((v * fI(u)) + ((1 - v) * gI(u)), u, ht(u), u, 0, 3, v, 0, 1) → Top
  20. Surface(fI(u), u, (v * ht(u)), u, 0, 3, v, 0, 1) → SurfaceFront
  21. Surface(gI(u), u, (v * ht(u)), u, 0, 3, v, 0, 1) → SurfaceBack