I have a question about an application of area and coarea formula and change of variables.
Let $D$ be a bounded and connected open subset of $\mathbb{R}^{d}$ with $C^{1}$-boundary. Define $D_{\epsilon}=\{x \in \bar{D}:d(x,\partial D)\leq \epsilon \}$. I want to show the following equation holds:
\begin{equation} \int_{D_{\epsilon}}\exp \left(-|x-y|^{2}\right)\,dy=\int_{0}^{\infty}\exp\left(-r^{2}\right)d\left\{ m(D_{\epsilon}\cap B(x,r))\right\}\cdots(1), \end{equation}
equivalently (by integration by parts formula), \begin{equation} \int_{D_{\epsilon}}\exp \left(-|x-y|^{2}\right)\,dy=\int_{0}^{\infty}2r\exp\left(-r^{2}\right) m(D_{\epsilon}\cap B(x,r))\,dr, \end{equation} where $B(x,r)$ and $m$ denote the open Ball of radius $r$ centerd at $x \in \mathbb{R}^{d}$ and $d$-dim Lebesgue measure, respectively. $d\left\{ m(D_{\epsilon}\cap B(x,r))\right\}$ is the Lebesgue stieltjes measure generated by the function $r \mapsto m(D_{\epsilon} \cap B(x,r))$ on $(0,\infty)$.
My attempt
Let $f$ be a $\mathbb{R}$-valued integrable function on $\mathbb{R}^{d}$. Note that the following equations hold: \begin{equation} \int_{\mathbb{R}^{n}}f\,dx=\int_{0}^{\infty} \frac{d}{dr} \left(\int_{B(0,r)} f\,dx \right)\,dr,\quad \frac{d}{dr}\int_{B(0,r)}f\,dx= \int_{\partial B(0,r)}f\,d\sigma, \end{equation} where $\sigma$ is the $(d-1)$ dim Hausdorff measure (surface measure). Using these equations, we have \begin{align} \int_{D_{\epsilon}}\exp \left(-|x-y|^{2}\right)\,dy&=\int_{0}^{\infty}\frac{d}{dr}\left( \int_{ B(0,r)}1_{D_{\epsilon}}(y)\exp\left(-|x-y|^{2} \right)\,dy\right)\,dr \\ &=\int_{0}^{\infty}\int_{\partial B(0,r)} 1_{D_{\epsilon}}(y) \exp\left(-|x-y|^{2} \right)\,d \sigma\,dr \\ &=\int_{0}^{\infty}\int_{\partial B(0,r) \cap D_{\epsilon}} \exp\left(-|x-y|^{2} \right)\,d \sigma\,dr \end{align} But I couldn't show \begin{equation} \exp\left(-r^{2}\right)d\left\{ m(D_{\epsilon}\cap B(x,r))\right\}=\int_{\partial B(0,r) \cap D_{\epsilon}} \exp\left(-|x-y|^{2} \right)\,d \sigma\,dr. \end{equation} If you know how to show the equation $(1)$, please let me know.
Thank you in advance.