Finding the Length of the Cable

We can use the arclength formula,

arclength = Integrate[Sqrt[1 + f'[x]^2]]

to find the length of the cable.


  Integrate[Evaluate[Sqrt[1 + D[catenary[x,t2], x]^2]],
  											{x,-50,50}]

Mathematica cannot integrate this function exactly. We must approximate the integral using NIntegrate. This command uses an algorithm called Gauss quadrature--but that is the subject of another lesson.


  NIntegrate[Evaluate[Sqrt[1 + D[catenary[x,t2], x]^2]],
  											{x,-50,50}]

So you will need an extra 2.6 meters of cable to span 100 meters.

Reference: This problem was adapted from Cheney and Kincaid, Introduction to Numerical Methods.

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