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Visualizing Volumes via Known Cross Sections

Visualizing Volumes via Known Cross Sections
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2016-03-18 06:56
PolygonSequenceSolid GeometryCalculus
In collections Calculus - Applications of Integrals - Areas and Volumes
This example provides an interactive visualization for computing volumes of solids using known cross sections. It demonstrates integral calculus concepts, particularly the method of slicing, and connects to solid geometry and continuity. The GeoGebra construction employs the Polygon command to create cross‑section shapes and the Sequence command to generate a series of parallel slices, with objects such as f, g, X, n, A, B, poly1, a, b, c, d, list1, e, h, list3, i, list2 enabling dynamic manipulation. Suitable for high‑school and undergraduate courses in calculus and mathematical modeling.

GeoGebra Commands & Parameters

  1. Polygon((X, f(X), 0), (X, f(X), f(X) - g(X)), (X, g(X), f(X) - g(X)), (X, g(X), 0)) → poly1, a, b, c, d
  2. Sequence(Polygon[(((k * X)) / n, f(((k * X)) / n), 0), (((k * X)) / n, f(((k * X)) / n), f(((k * X)) / n) - g(((k * X)) / n)), (((k * X)) / n, g(((k * X)) / n), f(((k * X)) / n) - g(((k * X)) / n)), (((k * X)) / n, g(((k * X)) / n), 0)], k, 1, n) → list1
  3. Sequence(CircumcircleArc[((k * X / n), f((k * X / n)), 0), ((k * X / n), (f((k * X / n)) + g((k * X / n))) / 2, (f((k * X / n)) - g((k * X / n))) / 2), ((k * X / n), g((k * X / n)), 0)], k, 1, n) → list3
  4. Sequence(Polygon[((k * X / n), f((k * X / n)), 0), ((k * X / n), (f((k * X / n)) + g(((k * X)) / n)) / 2, (sqrt(3) * (f((k * X / n)) - g((k * X / n))) / 2)), ((k * X / n), g((k * X / n)), 0)], k, 1, n) → list2