Differentiation of the Norm in $\mathbb{R}^n$

Question

I'm reading Evan's Partial Differential Equations. In the proof of the Theorem 1(iii) (Initial Value Problem of the Heat Equation on Page 48), the last step states $$\frac{C}{t^{n/2}}\int_{\mathbb{R}^n-B(x^0,\delta)}e^{-\frac{|y-x^0|^2}{16t}}dy=\frac{C}{t^{n/2}}\int_\delta^\infty e^{-\frac{r^2}{16t}}r^{n-1}dr$$ I think it's just a change of variable $r=|y-x^0|$. If so, $dr=\frac{y-x^0}{r}dy$. But I don't know how to proceed from here, and I don't see where that $r^{n-1}$ comes from. Can anyone help? Thanks in advance!