What course is this
(Calculus III does not teach this).
The problem asks by Green's Theorem as opposed to the standard method of calculating the the line integral.
You need to find,
Another notation for this (if that confuses you) is,
So the vector field is,
By Green's theorem,
So the double integral over that closed region is,
Can you take it from there?
(Hint: Divide your integral into two regions one is the square and the other is a semicircle.)
(For Engineering Students)
There happens to be a cool way to get that value if you ever taken an engineering course. You should recognize that is the first-moment. And hence you need to find the centroid of this shape. Which is simple for it is one of the standard shapes used.