I feel like this might almost be more of a physics than a math question, but I'll try to help by stating a few different things.
Generally, in terms of dimensional analysis, an $n$-fold integral over length-like coordinates will add $n$ length dimensions to the dimension of the integrand. This is why one calls triple integrals volume integrals and double integrals area integrals. But if the integrand itself is already of dimension $L^1$ (i.e., length), then a double integral of that integrand will have the dimensions of volume, $L^3$ (how one might calculate the volume of a cylinder, for example).
This dimensional analysis applies to integration with respect to non-length variables as well, but the interpretation may not always be as straightforward. Integration with respect to time $t$, for example, will typically be done on integrands of dimension $1/t$, though that is by no means a rigorous rule—it all depends on the mathematical or physical context. When calculating trajectories, for example, the integrand may be of dimension $L/t$ (velocity) and the integral with respect to $t$, leading to a length $L$ as a result. The key point, again, is that integration with respect to some variable increases the dimension of the result with respect to that variable by one.
This applies to line and surface integrals in the same way. General line and surface integrals are really just generalizations of 1D and 2D integrals to non-Euclidean (curved) spaces. It might be a good idea to look for pedagogical introductions into the basic concept behind integration on manifolds if you want to learn more about these generalizations.