Evaluating Integral for Region Bounded by x^2 - xy + y^2 = 2

[bobsmiters]

1. Find the area of the region bounded by x^2 - xy + y^2 = 2:
a)let x = au + bv, y= au - bv therefore, 3b^2v^2 + a^2u^2 = 2
b) Choose a and b such that u^2 + v^2 = 1, therefore, a = sqrt 2 & b = (sqrt 6)/3

c) Applying these results and changing variables into u and v, evaluate the integral //(x^2 - xy + y^2) dxdy, where the integral is bounded by the equation x^2 - xy + y^2 = 2.

For the part c) I have found the J(u,v) = 4(sqrt 3)/3, but in the examples I have I am supposed to follow this up with an integral and I am not sure what to do next. Are there any suggestions?

 

[HallsofIvy]

Integrate
$$\int\int J(u,v)dudv= \int_{u=-1}^1\int_{v= -\sqrt{2- 2u^2}/\sqrt{6}}^{\sqrt{2- 2u^2}/\sqrt{6}}\$$
over the uv-ellipse. Isn't that why you found J(u,v)?