# 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

• See this : youtube.com/watch?v=uXjQ8yc9Pdg – Victor Aug 5 '15 at 1:41
• I highly recommend Khan Academy for this topic. – Clarinetist Aug 5 '15 at 1:43
• I personally love this gif for getting an intuition of what the line integral is doing. – Hayden Aug 5 '15 at 1:47
• @Hayden wow that gif could have saved me a week of lectures in class – Elliot G Aug 5 '15 at 1:50
• The one I gave above was for a scalar field. This gif instead looks at vector fields, and is also really helpful (to me). @ElliotG It's amazing what a tiny gif like that can do for people's understanding! – Hayden Aug 5 '15 at 1:53

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.