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Volume of Revolution - Shell Method

Volume of Revolution - Shell Method
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2017-02-02 22:48
Solid GeometryCalculusIntegral
This example demonstrates how to calculate the volume of a solid of revolution using the shell method. Key mathematical concepts include integration and solid geometry. Using GeoGebra commands such as Point, Vector, Sequence, and Segment, a geometric model of the solid is constructed. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

  1. Point(xAxis) → Q
  2. Point(xAxis) → P
  3. Point(xAxis) → R
  4. Point(er_3) → M
  5. Vector(L, M) → u
  6. Sequence(Arc[x^(2) + y^(2) / b^(2) = (x(P) + (k * a))^(2), ((-(x(P) + (k * a))), 0), (x(P) + (k * a), 0)], k, 0, n - 1) → ellipse_3
  7. Segment((x(R) + a, 0), (x(R) + a, f_{oblique}(x(R)))) → s_1
  8. Segment(((-(x(R) + a)), 0), ((-(x(R) + a)), f_{oblique}(x(R)))) → s_2
  9. Segment(R, (x(R), f_{oblique}(x(R)))) → s_3
  10. Segment((-R), ((-x(R)), f_{oblique}(x(R)))) → s_4
  11. Sequence(Segment[(x(P) + (k * a), 0), (x(P) + (k * a), f_{oblique}(x(P) + (k * a)))], k, 0, n) → cut_1
  12. Sequence(Segment[(x(P) + (k * a) + a, ((1 - b) * f(x(P) + (k * a) + a))), (x(P) + (k * a) + a, f_{oblique}(x(P) + (k * a)))], k, 0, n - 1) → cut_2
  13. Sequence(Segment[((-(x(P) + (k * a))), 0), ((-(x(P) + (k * a))), f_{oblique}(x(P) + (k * a)))], k, 0, n) → cut_3
  14. Sequence(Segment[((-(x(P) + (k * a) + a)), f_{oblique}(x(P) + (k * a) + a)), ((-(x(P) + (k * a) + a)), f_{oblique}(x(P) + (k * a)))], k, 0, n - 1) → cut_4
  15. Sequence(x^(2) + (y - f_{oblique}(x(P) + (k * a)))^(2) / b^(2) = (x(P) + (k * a))^(2), k, 1, n) → ellipse_1
  16. Sequence(x^(2) + (y - f_{oblique}(x(P) + (k * a)))^(2) / b^(2) = (x(P) + (k * a) + a)^(2), k, 0, n - 1) → ellipse_2