Co-area formula and compositions

Question

I'm working on a problem and came across an integral which seems to involve the co-area formula and I wanted to see if what I'm doing is correct. Let's say we have an integrable function $F:\mathbb{R} \to \mathbb{R}$ and a Lipschitz function $g:\Omega \subseteq \mathbb{R}^n \to \mathbb{R}$, what I want to compute is $$\int_{\Omega} F(g(x)) dx$$ where $\Omega$ is some sufficiently "nice" region of $\mathbb{R}^n$. In practice what I'm interested in is very similar to a radial function, it's just a slightly more complicated function than $g(x) = r$. My intuition is that, in at least some sense, $$\int_{\Omega} F(g(x)) dx = \int_{\mathbb{R}}F(t)|g^{-1}(t)|dt $$ where $|g^{-1}(t)|$ is the surface area (Hausdorff measure?) of the level surface $\lbrace x \in \Omega \vert \hspace{0.2cm} g(x) = t \rbrace$. Can we use the co-area formula here in some way? It would be very useful if we could express this integral in terms of $|\nabla g(x)|$ but an ordinary change of variables doesn't seem to work and the co-area formula doesn't seem immediately applicable. Feels like this should be simple so maybe I'm just missing something. Grateful for any thoughts!