Let $X$ and $Y$ be Riemannian manifolds and consider a function \begin{align} f\colon X\times Y &\to \mathbb{R},\\ (x,y) &\mapsto f(x,y) \end{align} Now I have to integrate $f$ with respect to the Riemannian volume element of $X\times Y$. Is the infinitesimal volume element of $X\times Y$ the product of the infinitesimal volume elements of $X$ and $Y$?
My question could sound very naïve, but it is the first time that I work with Riemannian spaces and I couldn't find it in any textbooks.