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Line Integrals: Arc Length & Area of Fence

Line Integrals: Arc Length & Area of Fence
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2016-11-14 13:33
CircleCalculusIntegralGeometric ConstructionSlider Animation
This example demonstrates calculating arc length and area of a fence using line integrals. It involves core concepts like line integrals, definite integrals, and circle geometry. Key GeoGebra commands used include CurveCartesian for curve plotting, Derivative for slope calculation, Simplify for expression reduction, Integral for integration, Sum for summation, Textfield for annotations, and Sequence for creating sequences. Suitable for high school students and calculus learners to understand the application of line integrals in geometric problems.

GeoGebra Commands & Parameters

  1. CurveCartesian(pathx(i), pathy(i), pathz(i), i, ax, ay) → path
  2. Derivative(path) → Dpath
  3. Simplify(sqrt(x(Dpath(t))^(2) + y(Dpath(t))^(2) + z(Dpath(t))^(2))) → norm
  4. Integral(g, ax, ay) → Lineintegral
  5. Sum(Sequence[((Δt * f(x(path(k + Δt)), y(path(k + Δt)), z(path(k + Δt)))) * sqrt(x(Dpath(k + Δt))^(2) + y(Dpath(k + Δt))^(2) + z(Dpath(k + Δt))^(2))), k, ax, ay - Δt, Δt]) → Riemann
  6. Textfield(pathx) → InputBox1
  7. Textfield(pathy) → InputBox2
  8. Textfield(pathz) → InputBox3
  9. Textfield(f) → InputBox4
  10. Textfield(ax) → InputBox5
  11. Textfield(ay) → InputBox6
  12. CurveCartesian(pathx(i), pathy(i), i, ax, ay) → path2
  13. Derivative(path2) → Dpath2
  14. Simplify(sqrt(x(Dpath(t))^(2) + y(Dpath(t))^(2))) → norm2
  15. Integral(h, ax, ay) → Lineintegral2
  16. Sequence(path2(ax + (k * (ay - ax) / 100)), k, 0, 100, 1) → points
  17. Sequence(path2(ax + (k * (ay - ax) / 100)) + (0, 0, 0), k, 0, 100, 1) → pts
  18. Sequence((x(Element[points, k]), y(Element[points, k]), F(x(Element[points, k]), y(Element[points, k]))), k, 1, 100 + 1, 1) → list1
  19. Sequence(Polygon[Element[pts, k], Element[list1, k], Element[list1, k + 1], Element[pts, k + 1], Element[pts, k]], k, 1, 100) → list2
  20. CurveCartesian(x(path(k)), y(path(k)), F(x(path(k)), y(path(k))), k, ax, ay) → Surfacecurve
  21. Textfield(F) → InputBox7