Answer:
∫C (x - 3y) dx + (x y) dy = π/2
Explanation:
To apply Green's theorem, we need to first find the curl of the given vector field:
curl(F) = ∂Q/∂x - ∂P/∂y
where P = x - 3y and Q = xy
∂Q/∂x = y and ∂P/∂y = -3
So, curl(F) = y + 3
Now, we need to find the double integral of curl(F) over the region D between the given curves:
∬D curl(F) dA = ∫θ2θ1 ∫1/3R R (r sinθ + 3) r dr dθ (converting to polar coordinates)
where R = 3 and 1/3 respectively, and θ1 and θ2 are the limits of integration.
Solving the integral, we get:
∬D curl(F) dA = π/2
Thus, by Green's theorem, we have:
∫C (x - 3y) dx + (x y) dy = π/2
where C is the boundary of the region D, i.e., the curves x2y2 = 1 and x2y2 = 9. However, finding the line integral directly would be quite complicated, so it is much easier to use Green's theorem in this case.