# Graphs for which a calculus student can reasonably compute the arclength

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

$$∫ba√1+(f′(x))2dx\int_a^b\sqrt{1+\left(f'(x)\right)^2}\,\mathrm{d}x$$

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

• Letting $$ff$$ be linear is super easy, but then you don’t even need the formula.
• Taking $$ff$$ of the form $$(stuff)32(\text{stuff})^{\frac{3}{2}}$$ might work out nicely if $$stuff\text{stuff}$$ is chosen carefully.
• Calculating it for $$f(x)=√1−x2f(x) = \sqrt{1-x^2}$$ is alright if you remember that $$∫1x2+1dx\int\frac{1}{x^2+1}\,\mathrm{d}x$$ is $$arctan(x)+C\arctan(x)+C$$.
• Letting $$f(x)=ln(sec(x))f(x) = \ln(\sec(x))$$ results in $$∫sec(x)dx\int\sec(x)\,\mathrm{d}x$$, which classically sucks.

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

$$f(x)=12∫(g(x)−1g(x))dxf(x)=\frac12\int \left(g(x)-\frac1{g(x)}\right)\,\mathrm dx$$
for some suitably differentiable $$g(x)g(x)$$ over the desired integration interval for the arclength. The arclength over $$[a,b][a,b]$$ is then given by
$$12∫ba(g(x)+1g(x))dx\frac12\int_a^b\left(g(x)+\frac1{g(x)}\right)\,\mathrm dx$$
$$g(x)=x10g(x)=x^{10}$$ and $$g(x)=tanxg(x)=\tan x$$ are some of the example functions given in the article that are amenable to this recipe.