Electric potential of surface and volume charges?

The potential is not an absolute value, so what we have is the difference of potential between two points. That being said, the integral is not defined along the body producing the field, but along a path that connects two points, one of these points is a reference you adopt for the system, the other one is where you want to know the potential.

For example, for any finite object producing a field, you can define the reference point as the infinity with $V(infty) rightarrow 0$, so that the path of integration can be a line (or any other path) from infinity to the point you wanna know the potential, like: begin{equation} V(vec{r}) = - int_infty^{vec{r}} vec{E} cdot d vec{l} end{equation}

In the case of infinity objects $V(infty) neq 0$, so you have to choose another point of reference, and the above equation be begin{equation} V(vec{r}) - V( vec{r_0}) = - int_{vec{r_0}}^{vec{r}} vec{E} cdot d vec{l} end{equation} Just to answer the question, although I think you've realized, this is the equation of the potential for any body producing an electric field.