We can compute the integral directly: we have
Then the integral is
You could also take advantage of Stokes' theorem, which says the line integral of a vector field along a closed curve
is equal to the surface integral of the curl of
over any surface
that has
as its boundary.
In this case, the underlying field is
which has curl
We can parameterize by
with and
.
Note that when viewed from above, has negative orientation (a particle traveling on this path moves in a clockwise direction). Take the normal vector to
to be pointing downward, given by
Then the integral is
Both integrals are kind of tedious to compute, but personally I prefer the latter method. Either way, you end up with a value of .