compute following integral with respect to Hausdorff measure

Question

I would like to compute the following integral $$ \int_{x^2 + y^2 = r^2} \frac{1}{\sqrt{x^2 + y^2}} d\mathcal{H}(x, y) $$

My Attempt

I tried the following $$ \begin{align} \int_{x^2 + y^2 = r^2} \frac{1}{r} d\mathcal{H}(x, y) = \frac{1}{r}\mathcal{H}(\{(x, y)\in \mathbb{R}^2\, :\, x^2 + y^2 = r^2\}) = \frac{1}{r} \cdot 2 \pi r = 2 \pi \end{align} $$

Answer 1

For simple curves the Hausdorff measure $\mathcal{H}^1$ is just $\mathcal{L}^1$ the 1d Lebesgue measure. Your result is correct.