we will start with a two-dimensional curve C with parameterization,
The line integral of f with respect to x is,
The line integral of f with respect to y is,
Note that the only notational difference between these two and the line integral with respect to arc length (from the previous section) is the differential. These have a dx or dy while the line integral with respect to arc length has a ds. So when evaluating line integrals be careful to first note which differential you’ve got so you don’t work the wrong kind of line integral.
These two integral often appear together and so we have the following shorthand notation for these cases.
Let’s take a quick look at an example of this kind of line integral.
Example 1 Evaluate
Here is the parameterization of the curve.
The line integral is,