What quantity does a line integral represent? I'm currently trying to wrap my head around line integrals, Green's theorem, and vector fields and I'm having a bit of difficulty understanding what a line integral represents geometrically.
Is it basically the arc length of a curve, for a scalar field?
And then when you bring the concept into a vector field, then what does it represent?

For example if the integrand is a function of point mass, then the line integral over a curve will give you the mass of that curve. Also, when you integrate over a vector field, it physically represents the work done by the field on a particle that moves along a path.
A line integral can help you to calculate the area of a fence or curtain that's below the curve, in $\mathbb R^3.$
Rodolfo Llinás says that we learn when we put things into context. I believe him, and these interpretations of line integrals are very classical.
