Let $f:\mathbb R \to \mathbb R$ be a differentiable function, and consider the curve $\Gamma$ described by $y = f(x)$ over a closed interval $[a, b]$. As is well-known from basic Calculus, the arclength of $\Gamma$ can be calculated by computing the integral $$\int_a^b \sqrt{ 1 + \left( f'(x) \right)^2 } \, dx$$
My question concerns the integrand, considered as a function in its own right: what (if anything) is it called? That is, for a given function $f(x)$, is there a name for the function $$g(x) = \sqrt{ 1 + \left( f'(x) \right)^2 } ?$$
It feels like this is connected (sort of ) to a Jacobian but I can't quite articulate how. The best I have come up with is "arc length element function for $f$" but I'd like to know if there is a better name or more elegant way of describing this.