I have some questions about what the path an integral integrates over actually means.
I can one understand one interpretation of what is meant by $L$ in the following line integral.
$$ \Delta U = -\int_{\vec{s} \in L} \vec{F} \cdot \mathrm{d} \vec{s} $$
Clearly, $L$ is not a set of points because than the integral giving the expression for moving up from the surface of the earth to a point in the sky would be the same as moving from the point in the sky to the surface of the earth (instead of being the negation.)
One can rephrase the expression in terms of time as:
$$ \Delta U = -\int^b_a \vec{F} \cdot \frac{\mathrm{d} \vec{s}}{\mathrm{d} t} \, \mathrm{d} t $$
Under this interpretation L is a path travelled by the variable over time or a function of time to position.
However, I don't really understand how one can do the same thing for multiple integrals.
For a volume integral like:
$$ \int_{\vec{s} \in C} f\left(\vec{s}\right) \, \mathrm{d} C $$
I don't really understand how one would I lift the problem to time.
I think $C$ ends up being a function of time to a set of points but I'm not sure how to work with this to construct the proper integral.
The first integral features a path that is some vector valued function $$ s(t) : I \subseteq \mathbb{R} \to \mathbb{R}^3 $$ The second version is how you would actually evaluate it.
The third one I do not understand. I would expect the summation of volume elements in case of a volume integral.