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