5. Consider (y -x) = R ( xty ) ? dA where R is the region in the xy-plane bounded by the trapezoid with vertices (x =1,y=0), (x= 4, y= 0) , (x =0,y=1) and(x = 0,y =4) . The change of variables u = xty, v=y-x defines a transformation T(x, y) = (u, v) from the xy-plane to the uv-plane. (a) (16 pts) Sketch S and write S (in terms of u and v) using set-builder notation, where T: R->S. Use T to help you sketch S in the uv-plane by evaluating T at the vertices. 2 0 IN -2 -4 d(u, v) au au (b) (4 pts) Evaluate the determinant dx dy = a(x, y) av av ax ay (c) (18 pts) The Jacobian o(x, y ) d(u, v) (( -x ) "(u, v) a(x, y) use this fact to help evaluate R (xty ) 2 dA