Suppose $g: \mathbb{R}^{n} \to \mathbb{R}$ is $C^{1}$ and satisfies $\nabla g(x) \neq 0$ for every $x \in \mathbb{R}^{n}$. Let $E \in \mathbb{R}$ be fixed. I want to find an explicit expression for the integral: $$\int_{g(x) = E}f(x)dx,$$ provided the integral exists.
I sketched some calculations, but I am not completely sure if it is correct. There are some steps I am not still very convinced. My idea is the following. Let $\phi: \mathbb{R}^{n}\to U\subset \mathbb{R}^{n}$ be a change of variable transformation (diffeomorphism) given by: $$\phi(x) = (\phi_{1}(x),\phi_{2}(x),...,\phi_{n}(x)) \tag{1}\label{1}$$ where I choose $\phi_{1}(x) = g(x)$ and, for each $i=2,...,n$, $\phi_{i}: \mathbb{R}^{n} \to \mathbb{R}$ are $C^{1}$ functions such that the change of variables $\phi$ is orthogonal, in the sense that $\langle \nabla \phi_{i}(x),\nabla\phi_{j}(x)\rangle = 0$ for every $i\neq j$.
Consider the region: $$R = \{y = (y_{1},...,y_{n}) \in \mathbb{R}^{n}: y_{1} = E\}\tag{2}\label{2}$$ so we have: $$\{x \in \mathbb{R}^{n}: g(x) = E\} = \phi^{-1}(R) = \{x \in \mathbb{R}^{n}: \phi_{1}(x) = g(x) = E\}.$$ Using the change of variables formula, we obtain: $$\int_{g(x) = E}f(x) dx = \int_{R}f(\phi^{-1}(y))\frac{1}{|\det D\phi(\phi^{-1}(y))|}dy = \int_{\mathbb{R}^{n}}f(E,y_{2},...,y_{n})\frac{1}{\|\nabla g(E,y_{2},...,y_{n})\|}d\Sigma_{E}$$ where $\|x\| = \sqrt{x_{1}+\cdots +x_{n}^{2}}$ is the usual Euclidean norm and: $$d\Sigma_{E} = \frac{dy_{2}\cdots dy_{n}}{\prod_{i=2}^{n}\|(\nabla\phi_{i})(E,y_{2},...,y_{n})\|}$$
First of all, I would like to know if my reasoning and formula are correct. If so: Can I always find a change of variable $\phi$ which is orthogonal and has one of its components, say, $\phi_{1} = g$ as I did? I am not entirely convinced about it.
Look up the closely-related Co-Area Formula for integration over a family of submanifolds. You have outlined the key steps in a proof of it. Your ideas work locally ( if you abandon the assumption that all your local coordinate variables are orthogonal). The global proof can be patched together with a partition of unity.
Co-area formula in simple case
P.S. Your question is somewhat ambiguous because you have not specified explicitly that you are integrating over the hypersurface with respect to the canonically induced co-dimension one Hausdorff measure.