I read that we can do the following in the appendix of a book on seismic imaging: If $g(x) = 0$ represents the surface of some bounded domain then we can write $$ \int_V dV = \int_V dS dg/|\nabla g(x)| $$ where $dS$ is the differential surface area on the level surface(s) described by the condition $g(x) = \text{constant}$.
I don't see how this can be derived? In particular, how does the magnitude of the gradient of $g$ arise in the denominator?