For the vector field g? =(yexy+4cos(4x+y))i? +(xexy+cos(4x+y))j? g? =(yexy+4cos(4x+y))i? +(xexy+cos(4x+y))j? , find the line integral of g? g? along the curve cc from the origin along the xx-axis to the point (4,0)(4,0) and then counterclockwise around the circumference of the circle x2+y2=16x2+y2=16 to the point (4/2? ,4//2,4/2).

We're computing the line integral



It looks like the circular part of should be along the circle starting at (4,0) and terminating at .

Because integrating with respect to a parameterization seems like it would be a pain, let's check to see if is a conservative vector field. For this to be the case, if , then is conservative iff .

We have and . The corresponding partial derivatives are




and so the vector field is indeed conservative.

Now, we want to find a function such that . We have



Integrating with respect to yields




Differentiating with respect to gives






and so



Because is conservative, and a potential function exists, the line integral is path-independent and the fundamental theorem of calculus of line integrals applies, so we can evaluate the line integral by evaluating the potential function at the endpoints. We end up with