Definition:
The line integral (from a to b) of a vector field F(x) along a curve C(t) is defined as:
∫C F.ds = ∫ab F(C(t)).C'(t)dt
Those of you reading this are now split into three groups. One, the applied mathematicians, is thinking "Well, duh!". Another, the pure mathematicians, is ripping holes in what I said and arguing over whether any parametrisation of a curve will give the same answer (it does). The third group, the non-mathematicians, is scratching its collective head and wondering how the hell it got here from nodes about lesbian ninja monkeys. What follows is for the non-mathematicians:
What the hell is that curly S thing?
That, my friend, is an integral sign. It's one of those clever little ideas that separates adding up numbers from Real Mathematics. In its most basic form, it represents the area under a curve. Look at the purty picture:
f(x)
^
|
| _
| __/ \_
| / \
| __/ \___
| / | |\_
| _/ | X | \____ /
| / | | \/
|/ | |
0+-------+-----------+---------->x
0 a b
Apologies for the bad ASCII art, but that's supposed to be a graph of a function f(x). See that section with the big X in it? The area of that section is represented by ∫ab f(x) dx (pronounced: "the integral from a to b of f of x with respect to x" or "integral from a to b fof x d x"). There are ways of calculating these things, given a, b and f, but most mathematicians (and physicists and engineers) just look the things up in a table.
What the hell do I want to do that for?
Oh, lots of things. You see, lots of things in nature (and in mathematics) turn out to be related to each other by integrals. For example, if you're looking at a function v(t) which gives your speed at time t, then your distance travelled between time a and time b is ∫ab v(t) dt. If the current flowing through a wire at time t is I(t) then the total amount of charge transferred between time a and time b is ∫ab I(t) dt. Of course, there are many others, but I think you get the idea by now.
OK, so what's special about line integrals?
Well, ordinary integrals are fine when your function just takes in numbers and spits out numbers. Unfortunately, most things in nature don't work like that. Instead of scalar functions, we have vector functions. These are functions that take a position and give out a direction and a magnitude. For example, the wind can be viewed as a function which takes a point in the atmosphere and gives back a wind speed and direction. Now, with a little fiddling about, one can make the standard integral work for functions which take in numbers and give out vectors (direction/magnitude pairs). However, to deal with functions which take in vectors, we need to sit down and think for a bit. When we're finished, we've got three ideas (or more, but only three of them are much use). These are line integrals, surface integrals and volume integrals.
Yes, but what is it?
The idea with a line integral is that we take a curved path, straighten it out, and integrate as though this was our x-axis. Imagine taking a piece of string, and laying it through your space. With the string still in position, measure the value of the function at each point on the string. Then straighten the string out, and create a new function, based on the values on the string. This new function can be integrated, and we refer to this as the line integral.
Alright, I think I understand that. Why do I want to do this?
Lots of reasons. For example, the energy required to move along a given path is the integral of the force opposing you, taken along that path. In general, the line integral of any conservative field along a curve is equal to the difference in potential between the endpoints.
Wow, this stuff is really interesting! Where can I learn more?
A much more rigorously mathematical treatment of this should be in any first year undergraduate mathematics, physics or engineering course. If you want to learn for yourself, pick up a book on vector calculus (or calculus of multiple variables, or multivariate calculus, or whatever they feel like calling it). For an online resource, Eric Weisstein's World of Mathematics at http://mathworld.wolfram.com/ is excellent, but it's designed for someone who knows what they're doing already, and probably isn't great for a beginner.
This has been part of the Maths for the masses project