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Volume with Square Cross Sections

Volume with Square Cross Sections
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2016-03-19 11:36
Solid GeometryCalculusIntegralGeometric Construction
This example demonstrates calculating the volume of a solid of revolution using square cross sections. It involves core concepts like volume integration, solid geometry, and quadratic functions. Key GeoGebra commands used include Intersect, Segment, Locus, and Polygon. Suitable for high school and above students to explore the integration of volume calculus and solid geometry.

GeoGebra Commands & Parameters

  1. Intersect(log(x), 5 - x, 3, 4) → E
  2. Segment(A, B) → e
  3. Locus(C, s) → loc1
  4. If(togDF ≟ 1, "Show Instructions", "Show volume integral") → ToggleDirectionVolumeIntegral
  5. If(togS ≟ 1, "Hide surface", "Show surface") → ShowSurface
  6. Locus(D, s) → loc2
  7. Polygon(A, B, C, D) → Square, a, b, c, d
  8. If(togB ≟ 1, "HIde boundary square", "Show boundary square") → BoundarySquare
  9. Surface(u, (v * f(u)) + ((1 - v) * g(u)), 0, u, 1, 5, v, 0, 1) → Base
  10. Surface(u, (v * f(u)) + ((1 - v) * g(u)), g(u), u, 1, 5, v, 0, 1) → Top
  11. Surface(u, 0, (v * g(u)), u, 1, aa, v, 0, 1) → SurfaceFront_1
  12. Surface(u, 0, (v * g(u)), u, aa, 5, v, 0, 1) → SurfaceFront_2
  13. Surface(u, g(u), (v * (g(u) - f(u))), v, 0, 1, u, 1, aa) → SurfaceBack_1
  14. Surface(u, g(u), (v * (g(u) - f(u))), v, 0, 1, u, aa, 5) → SurfaceBack_2
  15. Sequence(Polygon[(1 + ((k + 1) * h), 0, 0), (1 + ((k + 1) * h), g(1 + ((k + p) * h)), 0), (1 + ((k + 1) * h), g(1 + ((k + p) * h)), g(1 + ((k + p) * h))), (1 + ((k + 1) * h), 0, g(1 + ((k + p) * h)))], k, 0, m - 1) → SlicesRight
  16. Sequence(Polygon[(1 + (k * h), 0, 0), (1 + (k * h), g(1 + ((k + p) * h)), 0), (1 + (k * h), g(1 + ((k + p) * h)), g(1 + ((k + p) * h))), (1 + (k * h), 0, g(1 + ((k + p) * h)))], k, 0, m - 1) → SlicesLeft
  17. Sequence(Surface[u, 0, (v * g(1 + ((k + p) * h))), u, 1 + (k * h), 1 + ((k + 1) * h), v, 0, 1], k, 0, m - 1) → SlicesFront
  18. Sequence(Surface[u, g(1 + ((k + p) * h)), (v * g(1 + ((k + p) * h))), u, 1 + (k * h), 1 + ((k + 1) * h), v, 0, 1], k, 0, m - 1) → SlicesBack
  19. Sequence(Surface[u, (v * g(1 + ((k + p) * h))), g(1 + ((k + p) * h)), u, 1 + (k * h), 1 + ((k + 1) * h), v, 0, 1], k, 0, m - 1) → SlicesTop
  20. Sequence(Polygon[(1 + (k * h), 0), (1 + (k * h), g(1 + ((k + p) * h))), (1 + ((k + 1) * h), g(1 + ((k + p) * h))), (1 + ((k + 1) * h), 0)], k, 0, m - 1) → SlicesBase
  21. Sequence(Area[Element[SlicesLeft, k]], k, 1, m) → Areas
  22. Polygon((bs, 0, 0), (bs, g(bs), 0), (bs, g(bs), g(bs)), (bs, 0, g(bs))) → BoundSquare, i, j, k, l