How to simplify the following integral using spherical coordinates \begin{align} \int_{\|x\| \le R} e^{x \cdot y} dx \end{align} where $x, y\in \mathbb{R}^n$.
I know how to do this with cylindrical and spherical coordinates in $n=2$ using the formula $ x \cdot y=\|x\| \|y\| \cos(\theta)$.
Can someone explain how to simplify this integral by using spherical coordinates especially for larger $n$ (i.e. $n>3$ ).
I am specifically interested in how to properly do the change of coordinate systems. Once this is done I can do the integral myself.
You may assume ${\bf y}=\eta\,{\bf e}_n$ with $\eta>0$. Then by Fubini's theorem we have $$\int_{B_n(R)}e^{{\bf x}\cdot{\bf y}}\>{\rm d}({\bf x})=\int_{-R}^R e^{\eta x_n} \int_{B_{n-1}\bigl(\sqrt{R^2-x_n^2}\bigr)}{\rm d}({\bf x}')\>dx_n\ .$$ Here the inner integral is just the volume of $B_{n-1}\bigl(\sqrt{R^2-x_n^2}\bigr)$.