TITLE: Finding the inverse of the arc length function
QUESTION [13 upvotes]: I'm just a simple high school math student, so please don't eat me =)
In my calculus text, I have the formula:
$$L(x) = \int_{c}^{x} \sqrt{[f'(t)]^2 + 1}\,dt$$
Where $L(x)$ is the arc length of a curve $f(x)$ from $c$ to $x$.
How can I invert this function so that I can find valid values of $x$ to satisfy a given arc length? Something like $L^{-1}(x)$.

REPLY [5 votes]: You can do it by a differential equation without getting $L(x)$ explicitly.  If $\frac{dL}{dx} = \sqrt{1 + f'(x)^2}$, then
 $\frac{dx}{dL} = \frac{1}{\sqrt{1 + f'(x)^2}}$.  Numerical methods can be used to solve this differential equation.