What exactly IS a line integral?

As what happens in many math courses, a topic is learned without truly learning what one is doing. For me, this is line integrals. I can do them well, I just never truly learned what exactly I was doing. Can anyone give me (in layman's terms, something extremely basic) a good definition and example of what line integrals are truly evaluating and why we do it?

Thank you

Solution 1:

From my prospective, the concept of line (curve, path) integral is not much different from the concept of regular one dimensional (definite) Riemann integral.

One way to interpret the Riemann integral is to perceive it as the area under the curve. Very often Riemann integral is introduced via Riemann sums, which plays well with its "area-under-the-curve" interpretation.

You can imagine the process of computing definite integral of a (WLOG positive) function $f: \Bbb R \to \Bbb R$ on the interval $[a,b]\ni \Bbb R$ via Riemann sums as an iterative process: starting from $a$ and taking infinitesimally small steps towards $b$, we multiply the length of the step by the value of $f$ on this interval. Basically, on each step we compute the area an of infinitesimally small rectangle restricted by the value $f$ from above, and by the $x$ axis from below. Then we sum these tiny rectangle areas over the whole $[a,b]$ interval and thus approximate the total area under $f$: [

Similarly, one can define path integral of $f$ over a smooth curve $C$ via Riemann sums. The construction process is merely identical: we start from one side of the curve and take infinitely small steps towards the other. On each step we consider rectangle bounded by $f$ from above and by $C$ from below. Within each step the "base of the rectangle" part of $C$ is, essentially, straight, so we can compute the area of the rectangle just like in case of definite integral on a straight line. Summing areas of all rectangles will give us the "area" under the function $f$ on this path.

In essence, one can say that the line integral of $f$ over curve $C$ shows us the area which would be restricted by $f$ if we "straightened" and "stretched" the domain curve $C$ into a straight line.