I just (barely) passed what is known as my university's hardest first year physics course: electricity and magnetism. We didn't use differential equations, but we had the same hand wave "just think about it" relationship with line integrals. The way my professor explained it was "we're engineers, not mathematicians, so we actually know what we're talking about".

What he meant by this is that as engineers, we need to have intuition as to what an integral actually is rather than knowing how to prove things with them. To an extent, this is a good way to go about it. Really it's not at all useful to understand things like delta-epsilon as an engineer because those concepts really have no application. It's much more useful to just gain an intuition as to what you're actually doing when you're taking a line integral, and applying that to, for example, calculating an electric field strength as a result of an object.

Physics teaches physics, not math.