I'm confused with the volume of an n-dimensional Riemannian manifold with a scaled metric. Specifically, let $M$ be a Riemannian manifold with Riemannian metric $g$ and its volume denoted by $vol_g(M)$. If we endow $M$ with another metric
$$\widetilde{g}=\lambda g,\qquad \text{here $\lambda$ is a positive smooth function on $M$}$$
then, how about the volume $vol_{\widetilde{g}}(M)$ of with respect to $\widetilde{g}$? I know that the volume form
$$dv_{g}=\sqrt{\det\ g}\ dx^1\wedge\cdots \wedge dx^n .$$
So I guess that
$$dv_{\widetilde{g}}=\lambda^{n/2}dv_{g}, $$
and therefore
$$vol_{\widetilde{g}}(M)=\lambda^{n/2}vol_g(M) .$$
Do you think it is right? If not, how do we to calculate the volume $vol_{\widetilde{g}}(M)$ and figure out the relationship with $vol_g(M)$?
Thanks in advance!