Graphs for which a calculus student can reasonably compute the arclength

Given a differentiable real-valued function f, the arclength of its graph on [a,b] is given by


For many choices of f this can be a tricky integral to evaluate, especially for calculus students first learning integration. I’ve found a few choices of f that make the computation pretty easy:

  • Letting f be linear is super easy, but then you don’t even need the formula.
  • Taking f of the form (stuff)32 might work out nicely if stuff is chosen carefully.
  • Calculating it for f(x)=1x2 is alright if you remember that 1x2+1dx is arctan(x)+C.
  • Letting f(x)=ln(sec(x)) results in sec(x)dx, which classically sucks.

But it looks like most choices of f suggest at least a trig substitution f(x)tan(θ), and will be computationally intensive, and unreasonable to ask a student to do. Are there other examples of a function f such that computing the arclength of the graph of f won’t be too arduous to ask a calculus student to do?


Ferdinands, in his short note “Finding Curves with Computable Arc Length”, also comments on the difficulty of coming up with suitable examples of curves with easily-computable arclengths. In particular, he gives a simple recipe for coming up with examples: let


for some suitably differentiable g(x) over the desired integration interval for the arclength. The arclength over [a,b] is then given by


g(x)=x10 and g(x)=tanx are some of the example functions given in the article that are amenable to this recipe.

Source : Link , Question Author : Mike Pierce , Answer Author : J. M. ain’t a mathematician

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