Is there an intuitive, physical way to see the line integral?

Question

The way I am thinking of a line integral intuitively is if I had a copper wire, that was poorly made and had different densities throughout the wire, then I could chop the wire up into several pieces, and use the density at that piece and the length of the copper wire to find the mass of the wire at that point. This is what the line integral is calculating to me. Am I right in thinking this? Is there a better way to visualize the line integral? I am already familiar with the work and physics examples so I was trying to come up with a way to visualize the integral in another way.