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Volume of Solids of Revolution by Claudio

Volume of Solids of Revolution by Claudio
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2015-04-23 15:50
FunctionSolid GeometryCalculusIntegralGeometric Construction
This example demonstrates the calculation of the volume of solids formed by revolving 2D shapes around an axis, utilizing integral calculus. It involves visualizing functions, generating sequences, and plotting surfaces. Key math concepts include rotation, integration, and geometric transformations. The construction uses GeoGebra commands like Element, Sequence, Surface, Integral, and LeftSum. Ideal for high school and college students learning calculus and geometry applications.

GeoGebra Commands & Parameters

  1. Element(listaf, function) → f
  2. Sequence((i / n * b), i, 0, n) → lista1
  3. Sequence(f(Element[lista1, i]), i, 1, n + 1) → lista2
  4. Sequence(Cylinder[(Element[lista1, i], 0, 0), (Element[lista1, i + 1], 0, 0), f(Element[lista1, i])], i, 1, n + 1) → lista3
  5. Surface(t, (f(t) * cos(r)), (f(t) * sin(r)), t, 0, b, r, 0, 6.283185307179586) → a
  6. Sequence((Element[lista1, i], 0), i, 1, n + 1) → lista4
  7. Sequence((Element[lista1, i], Element[lista2, i]), i, 1, n) → lista5
  8. Sequence(Polygon[Element[lista4, i], Element[lista5, i], (Element[lista1, i + 1], Element[lista2, i]), Element[lista4, i + 1]], i, 1, n) → lista6
  9. Integral((π * f(x)^(2)), 0, b) → Volume
  10. LeftSum((π * f(x)^(2)), 0, b, n) → LeftSumAppx
  11. Sequence(((π * Element[lista2, i]^(2)) * b / n), i, 1, n) → lista7
  12. Sum(lista7) → g
  13. Line(A, B) → c
  14. Segment(A, B) → d
  15. Intersect(d, xAxis) → C