I have just begun exploring the topic of line integrals for my Calculus 3 class. Although I can perform the calculation properly, I don't understand the physical significance of what I've just calculated.

For example, I know that when I calculate the integral of a function which is defined along x, the physical significance is that I have calculated the area under the curve.

What then, would be the equivalent, in plain English, of the result that is obtained when calculating the integral of a function along a curve? It is tempting to guess that one has obtained the area between the function and the curve, but I don't think so, because that would just be the difference of two regular integrals.

I have come across certain analogies of the curve C being the path of a particle being pushed by a variable force F which is the function to be integrated over C. However, I'm still having a bit of trouble visualizing all of this. Can anyone provide some insight or a good way to conceptualize what exactly is happening when one computes a line integral?

For example, I know that when I calculate the integral of a function which is defined along x, the physical significance is that I have calculated the area under the curve.

What then, would be the equivalent, in plain English, of the result that is obtained when calculating the integral of a function along a curve? It is tempting to guess that one has obtained the area between the function and the curve, but I don't think so, because that would just be the difference of two regular integrals.

I have come across certain analogies of the curve C being the path of a particle being pushed by a variable force F which is the function to be integrated over C. However, I'm still having a bit of trouble visualizing all of this. Can anyone provide some insight or a good way to conceptualize what exactly is happening when one computes a line integral?

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