What is the geometric meaning of a line integral?

[amcavoy]

I understand how to evaluate a line integral, but I don't know what it represents geometrically. Say you have $$\int_{C} x^4yd\mathbf{s}$$. What does this mean geometrically? I can see that $$\int d\mathbf{s}$$ is the length of the arc (am I correct?), but I just can't seem to figure out what $$\int_{C}f(\mathbf{r}(t))||\mathbf{r}'(t)||dt$$ means.

Thanks for your help.

 

[Gza]

I usually like to think of line integrals as computing the area of a fence, with the fence's shape being defined by r(t). The height of the fence is simply the function you compute the line integral over.

 

[whozum]

Someone once told me that the curve in a line integral (ds' limits) is similar to the x-axis in the standard integral. So for example in a regular integral, each dx segment is a section of the x axis, but in a line integral each ds segment is a section of the curve you are integrating.

 

[mathwonk]

remember an integral is not area, but merely a sum of things each multiplied by a length of a small segment in the domain. i.e. it is a sum of products, where one factor is an interval in the domain nand the other factor is a value of the function being integrated.


If we think of the values of the function as heights of point on a graph, and the domain is the x axis, then this product, an x interval times a heignt, is naturally an area between the x-axis and the graph.


but an integral is also a change in the value of the antiderivative, i,e, the quantity represented by the integral. If this is naturally an area then it isa change in area, but it could be something else. e.g. if we are integrating the velocity function then the integral is the displacement function, i.e. change in position.


If we are integrating the area of a slice function for a solid, then the integral is the change in the volume between two slices.


if we integrate something like dtheta, over an arc, we are emasuring the change in theta as we move along the arc, i.e. the angle swept out by the points of the arc wrt the origin.

not all path integrals depende only on endpoints, so not all path integrals can be thought of as a simple change from one end to the other, but the most important ones can. i.e. anyone form whose curl is zero is a total differential of some quantity, and its path integral is the just the change in that quantity whatever it is from one end to the other of that path.


even a differential, which is not path independent, is a change in some quantity along that one path.


so remember integrals merely express a relationship between two quantities. If the integrand is height then the integralk is area, but not otherwise.


we are so used to only interpreting integrals one way it becomes confusing in path integration, where the most sueful integrands are not heights at all, but some otther interesting quantities, like rate of change of angles, or change in arc length, or force.

so try to inhterpret eaxh iontegral according to what the integrand means.

but in general you are measuring the change in some quantity along the arc, but it isn't always area in any natural way.


my favorite path integral is the winding number integral, actually just the angle change integral divided by 2pi.