Let the equation $f(x,y)=0$ describe some simple curve in the $x,y$-plane. I had the expectation that the length of this curve between $x_1$ and $x_2$ could be written as $$ \int_{x_1}^{x_2} dx\int_{-\infty}^\infty dy \delta(f(x,y)),$$ in the sense that the $\delta$ would pick up the points in the curve.
This works for a circumference, $f(x,y)=\sqrt{x^2+y^2}-R$ since $$\int_{-\infty}^{\infty} dx\int_{-\infty}^\infty dy \delta(\sqrt{x^2+y^2}-R)=2\pi \int_0^\infty rdr\delta(r-R)=2\pi R.$$
Does it work in general? The length should be $$ \int_{x_1}^{x_2} dx \sqrt{1+\left(\frac{dy}{dx}\right)^2},$$ I don't see that this equals my first integral.
