I am new to SE, so hopefully, I am asking this in the right format. I have the following integral, which comes out of a heat equation problem:
$$\int_{\mathbb{R}^3} e^{-{|x-y|^2/(4Dt)}}g(|y|) dy$$
where $x,y \in \mathbb{R}^3$, and $g(|y|) = c$ for $|y|\leq \alpha$ and $g(y)=0$ for $|y|>\alpha$. It seems logical to compute this integral in spherical coordinates, given that $g(|y|)$ restricts the integral to a sphere of radius $\alpha$, but the integrand does not just depend on $|y|$ in the usual way. I can probably compute the integral in Cartesian coordinates, but it seems like there should be a better method. Any suggestions are greatly appreciated!