  The art of analysis involves the subtle combination of approximation,
inequalities, and geometric intuition as well as being able to work at
different scales. With this subtlety in mind, we present this paper in a manner
designed for wide accessibility for both advanced undergraduate students and
graduate students. The main results include a singular integral for measuring
the level sets of a $C^{1,1}$ function mapping from $\mathbb{R}^n$ to
$\mathbb{R}$, that is, one whose derivative is Lipschitz continuous. We extend
this to measure embedded submanifolds in $\mathbb{R}^2$ that are merely $C^1$
using the distance function and provide an example showing that the measure
does not hold for general rectifiable boundaries.
