Not a big jump, if you conceptually stick to the 'area under curve' definition.

Analytically computing surface/volume integrals, yes, that is a big jump. But I don't understand why students need to memorize N algorithms and M substitution rules for that.

What is the straightforward "area under the curve" definition of a line integral?

Wikipedia has a nice animation for this.

https://en.wikipedia.org/wiki/Line_integral#/media/File:Line...

Okay. I am used to this being called a path integral, and the phrase line integral being reserved for the integral of a vector field over a curve. This does not, as far as I know, admit a straightforward definition as an area under a curve.

If you replace area under curve with volume under surface, it seems to me that the same intuition holds.

Sure, and AI isn't a big jump from statistics.

I would say AI:Statistics :: Engineering:Physics .

Sometimes it is a big jump, sometimes it is all common sense.