I would like to know if there exists some useful formulas that let us compute the volume form for a warped product, that is a $n$-dimensional Riemannian manifold of the form $dr^2 + \phi(r)^2 g_{N^{n-1}}$, where $N^{n-1}$ is a $n-1$-dimensional riemannian manifold and $g_{N^{n-1}}$ is the metric induced on it.
Also the case where $N^{n-1}$ is the hypershere in $\mathbb{R}^n$ would be useful: in this case computations could be carried out by direct computation, but I find it awkward.
I thank you in advance for any suggestion, Mattia.