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Demonstration of Volume of Solids of Revolution

Demonstration of Volume of Solids of Revolution
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2018-12-12 10:34
Solid GeometryCalculusIntegral
This example demonstrates how to generate a three-dimensional solid of revolution by rotating a two-dimensional shape and calculating its volume. The core mathematical concepts involved include the formula for the volume of solids of revolution, integral calculus, function graphs, and analytical geometry. Key construction elements from the GeoGebra command summary (Element, Sequence, Surface, Integral, LeftSum, Sum, Line, Segment, Intersect) are utilized. It is suitable for high school and above students studying calculus and solid geometry to understand the calculation methods for volumes of solids of revolution.

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