From my understanding the geometrical interpretation of a line integral simply is the area under a curve in a multidimensional space. I think this understanding of the geometrical interpretation corresponds to the line integral with respect to arc length, is this correct? What is the geometrical interpretation of a line integral with respect to a variable? If they have the same geometrical interpretation, then what is the difference between the two?
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There's a geometric interpretation that the answers there don't address. I'll try to find a better similar question. – Mark S. May 21 '21 at 09:28
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Tony Piccolo's answer to "What is the difference between a line integral with respect to x or y and a Riemann integral with respect to x or y?" contains a different sort of interpretation that might help. – Mark S. May 21 '21 at 09:37
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bfhaha's answer to "Interpreting Line Integrals with respect to $x$ or $y$" has a useful picture. And Interpretation of a line integral with respect to x or y . is another related question on MathSE (albeit with no answers) – Mark S. May 21 '21 at 11:29