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Arc Length of Parametric Curves

Arc Length of Parametric Curves
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2021-10-16 09:18
Coordinate GeometryCalculusIntegralParametric EquationGraphing
In collections MATH1052-edX
This GeoGebra applet demonstrates how to dynamically calculate the arc length of parametric curves. By inputting the upper and lower bounds of the parametric equations, the CurveCartesian command draws the curve while Zip and If commands perform numerical integration approximation, visually showing the discretized solution of the arc length formula. The applet includes four input boxes to control the parameter range and sampling points, updating the curve shape and arc length approximation in real time. Core mathematical concepts include derivatives of parametric equations, the arc length integral formula, and numerical approximation methods. Suitable for higher mathematics and coordinate geometry courses, or as an interactive demonstration tool for learning numerical computation methods.

GeoGebra Commands & Parameters

  1. CurveCartesian(f(u), g(u), u, min, t) → crv
  2. RemoveUndefined(Zip[min + (k * (t - min) / n), k, 0…n]) → Ls
  3. Zip(crv(k), k, Ls) → LP
  4. If(min ≟ max, PolyLine[Join[{LP, First[LP]}]], PolyLine[LP]) → appr
  5. Textfield(f) → InputBox1
  6. Textfield(g) → InputBox2
  7. Textfield(min) → InputBox3
  8. Textfield(max) → InputBox4

AI Drawing Prompt Reference

Draw a parametric curve in the plane, with x and y coordinates defined by two input functions. Show a set of discrete sample points evenly spaced along the parameter interval, with the count controlled by a sampling number. Connect these points with a polyline to approximate the curve's arc length piecewise linearly; if the lower and upper parameter bounds are equal, close the polyline. Include four input boxes for adjusting the two coordinate functions, the lower and upper bounds, and the sample count. As inputs change, update the curve and polyline approximation in real time to illustrate numerical integration of the arc length formula.
The prompt above was compiled from the original drawing commands and is for reference only.