General formula for volume integral of scalar field?

I am looking to derive general formulas for the electric fields generated by general charged objects given their charge densities. For linear and surface charge densities I have been able to derive the following expressions for a linear charge density $$\lambda(\vec{r})$$ and surface charge density $$\eta(\vec{r})$$, respectively:

$$\vec{E}=\frac{1}{4\pi\varepsilon_0}\int_C\frac{\lambda(\vec{r})}{\Vert\mathbf x_2-\mathbf x_1\Vert^3}(\mathbf x_2-\mathbf x_1)\,ds$$ $$\vec{E}=\frac{1}{4\pi\varepsilon_0}\iint_S\frac{\eta(\vec{r})}{\Vert\mathbf x_2-\mathbf x_1\Vert^3}(\mathbf x_2-\mathbf x_1)\,dS$$

I am able to compute these given that I am able to find a parameterization of the curve $$C$$ or the surface $$S$$, using the following formulas: $$\int_C f\,ds=\int_a^b f\big(\vec{r}(t)\big)\Vert\vec{r}'(t)\Vert\,dt$$ $$\iint_S f\,dS=\int_c^d\int_a^b f\big(\vec{r}(u,v)\big)\Vert\partial_u\vec{r}\times\partial_v\vec{r}\Vert\,du\,dv$$

However, I am uncertain how to do this in the case with a volume charge density $$\rho(\vec r)$$. I do not know a formula to simplify the volume integral. My guess would be maybe something like: $$\iiint_Rf\,dV=\int_\alpha^\beta\int_\gamma^\delta\int_\epsilon^\zeta f\big(\vec r(u,v,w)\big)\Vert\partial_u\vec r\times\partial_v\vec{r}\times\partial_w\vec{r}\Vert\,du\,dv\,dw$$

But I do not know a method to derive this. Is there a method/formula for computing these types of integrals given a parameterization of the region $$V\subset\mathbf{R}^3$$?

For a volume integral over an open set $$U \subseteq \mathbb{R}^3$$, the expression is pretty straightforward. Since this question is physics related, I will introduce coordinates into the expression in a very explicit way. Let $$\phi: U \to \mathbb{R}^3$$ denote the coordinate map. The volume integral is then given by
$$\int\int\int_UfdV := \int \int \int_{\phi(U)}(f \circ \phi^{-1})Jdq_1dq_2dq_3$$
where $$J$$ denotes the Jacobian determinant. Integration on the right-hand-side is a triple integral on the image of $$U$$ under $$\phi$$.