line integrals are basically integrals taken over a path in space. if you have a path mapped out in space and its piecewise smooth you can take the line integral. its pretty analogous to the plain old definite integrals youd do in an elementary calculus course (in fact it can be reduced to this type of integral). the only difference is that the line integral you divide subarcs and both variables are correlated to a parameter. geometrically you can interpret the meaning of a line integral like this: say you have a styrofoam cup with the bottom cutoff and you rest it on a table. now start cutting from the top of the cup up and down until it looks like you have sort of hills turning about in a circle. line integral would be like the area of one side of the cup. in this case the height of the hills just correlate to some crazy function and the path is the circle that the cup traces out. Physically you can interpret the line integral in terms of work. Say youre moving about some curve in space your direction of course is the tangent to each point in space. From physics its known that work is how much force is being done along some direction so naturally youd dot your force function with your direction and then integrate that with respect to an infinitesimal along the curve. so the integral turns out being integral F * T ds along a curve C where T is the tangential component of a vector. this can be written as F * dr
