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Visualizing Volumes by Known Cross Sections by Joy

Visualizing Volumes by Known Cross Sections by Joy
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2019-02-23 01:08
PolygonSequenceCalculusIntegral
This GeoGebra example dynamically visualizes the computation of volumes using known cross sections. By integrating cross-sectional area functions along an axis, it demonstrates the definite integral's geometric application in volume calculation. The construction employs Polygon and Sequence commands along with key objects like functions f, g, variables X, n, and points A, B to generate adaptive cross-sectional polygons. It serves as an interactive tool for calculus students and educators to explore the relationship between two-dimensional cross sections and three-dimensional volume, reinforcing the conceptual foundation of definite integrals in solid geometry.

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