Final answer:
To evaluate the given double integral over region R using the change of variables u = x-y and v = x+y, one must find the new integration limits and the Jacobian, which is 1/2 for this transformation, and then solve the integral.
Step-by-step explanation:
To evaluate the double integral ∬(x-y)²cos²(x+y) dx dy over the region R, which is bounded by the square with vertices (0,1), (1,2), (2,1), and (1,0), we use the suggested change of variable: u = x-y, v = x+y. This requires us to also determine the new region of integration in the uv-plane, as well as the Jacobian of the transformation, which in this case will be |∂(x,y)/∂(u,v)| = 1/2. After setting up the integral in terms of u and v, we need to find the limits for u and v that correspond to the original square region. The new bounds will be straight lines in the uv-plane, and we integrate first with respect to u, then v, or vice versa depending on the setup. The resulting integral will be simpler and can be solved using standard techniques of integration.