Suppose we have a $n$ dimensional manifold $M$, with local coordinates $\Phi:U \to V$. Further suppose that $\Sigma \subset M$ is an $(n-1)$-dimensional embedded submanifold. I understand that $$ \int_U dV_g = \int_V \sqrt{|g|} d\mathcal{H}^n,$$ where $\mathcal{H}^n$ denotes the Haussdorf (Euclidean) measure. How do I represent the area of the submanifold in the specified coordinates? I.e. I am looking for the function $h$ such that $$ \int_{U \cap \Sigma} dV_{i^*g} = \int_{V \cap \Phi(\Sigma)} h(x) d\mathcal{H}^{n-1}.$$
If $\Phi$ was a diffeomorphism between Euclidean spaces, then we have the formula $$\mathcal{H}^{n-1}(\Phi(\Sigma)) = \int_\Sigma \sqrt{\det [(\nabla^\Sigma \Phi)^T (\nabla^\Sigma \Phi)] } \mathcal{H}^{n-1}$$ where $\nabla^\Sigma \Phi$ denotes the tangential derivative. My guess is that if we want to generalize to the manifold setting, the formula would become something like $$ \int_{U \cap \Sigma} dV_{i^*g} = \int_{V \cap \Phi(\Sigma)} \sqrt{\det [(\nabla^\Sigma \Phi^{-1})^T G (\nabla^\Sigma \Phi^{-1})] } d\mathcal{H}^{n-1},$$ but this is beyond my powers to prove.
I tried reading Lee's Smooth Manifolds (chapter 15), but this ultimate proved unhelpful, as I do not know how to represent the induced volume form $i^*(N|\omega_g)$ in terms of local coordinates. Is there a resource that treats the area of submanifolds in better detail?