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The art of analysis involves the subtle combination of approximation,inequalities, and geometric intuition as well as being able to work atdifferent scales. With this subtlety in mind, we present this paper in a mannerdesigned for wide accessibility for both advanced undergraduate students andgraduate students. The main results include a singular integral for measuringthe 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 extendthis to measure embedded submanifolds in $\mathbb{R}^2$ that are merely $C^1$using the distance function and provide an example showing that the measuredoes not hold for general rectifiable boundaries.

A Singular Integral Measure for $C^{1,1}$ and $C^1$ Boundaries