So this should be explained to the students instead of letting them feeling around in the dark confused.
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.
Another way to see it is you cannot start a turbine by dumping fuel in and igniting it. The turbine has to be spun up first, usually with an electric motor. The Me262 engines had an ingenious tiny gas engine in the nacelle which served that purpose, complete with a little handle to start it!
Next time you're on a jet, watch them do the engine start. You can see it slowly gaining speed, then suddenly it dramatically speeds up - that's when the fuel gets squirted in and ignited.
Anyhow, jet engines look deceptively simple. But they are real masterpieces of engineering.
Michael Faraday was brilliant and driven, but his intuition did not come from understanding the math as he was famously math-illiterate.
On a personal note, the subject I had the most trouble with was classic electromagnetism (calc-based freshman physics): It was the first time I hit a wall/limit on what I could grasp.
The wall was because I could never couldn’t develop an intuition for the physics behind it.
On the off chance that you can understand German, check out chapter 9.1.1 Fluidvolumen und Divergenz (until and including chapter 11 Erhaltung der Energie), in the book Einführung Theoretische Meteorologie by Michael Hantel (ISBN 978-3-8274-3055-7). It even uses a car analogy ;)
You will understand fluid dynamics, and by extension electrodynamics, like you never did before. And it will teach you the physical intuition first.
This book was such a lucky find for me ten years ago.
Equation (11.3) is the fluid analogon to the continuity equation (in electrodynamics that is: ∂/∂t ρ + ∇⃗⋅J⃗ = 0)--and you can see how, physically, you get to it, and what the parts and the whole mean. Lots of diagrams and geometry and simple analogies make that take about 30 pages.