Consider Green's Theorem
For F = (x-y) i +(x +y) j
d(x+y)/dx = 1 d(x-y)/dy = -1
The line integral then is 2times the double integral over R , or twice the area of the region enclosed by your ellipse
It is easier to compute this with the line integral than with a double integral.
so the area enclosed is 1/2 the line integral
parameterize C with x = acos(t) y = bsin(t)
and now you are good to go.
Remember you should get pi*ab