2 y+ x 1. Consider the integral cos dA where R is the region in the xy-plane bounded by the trapezoid with L2 y-x R vertices: (0,2), (0,4),(-4,0), (-8,0). (a) (10 pts) Make a sketch of R, label the boundary lines in terms of x and y. 8 -7-6-5 -4-3-2 12 (b) (10 pts) The change of variables u = 2y- x, v = 2y + x defines a transformation T(x, y) =(u, v) from the xy- plane to the uv-plane. Solve for x and y in terms of u and v , then you will have the transformation T-'(u, v) = (x,y) from the uv-plane to the xy-plane.a(x, y) (c) (10 pts) Find the absolute value of the Jacobian, i.e., find au, v)(d) (10 pts) Make a sketch of the region S in the uv-plane such that T |R = S where R is the region from part (a) and T is the transformation in part (b) and write S using set-builder notation. 8 0- 8 12 -4 - 8(e) (16 pts) Use the Change of Variables Theorem (Theorem 14.5) to compute the exact value of If cos 2 y+ x dA by hand (no software) showing each step and simplify. 2 y -x R