1. Consider the region in the xy-plane given by: R = {(x, y) : 0

1. Consider the region in the xy-plane given by: R = {(x,y) : 0 < x < 2,0 < y 3+ 3x2} . (a) [1 mark]. Sketch the region R. (b) [2 marks]. Evaluate the integral 2y dx dy . We now introduce a new coordinate system, the vw-plane, which is related to the xy-plane by the change of coordinates formula: (x, y) = (v, w(l + UP)) . (c) [2 marks]. Calculate the Jacobian determinant for this change of coordinates; recall this is given by: (x, y) x/v x/w = det y/v y/w (v, w) (d) [2 marks]. Show the region R of the xy-plane corresponds to the region S of the vw-plane, where (e) [1 mark]. Use parts (c) and (d) to rewrite the integral in part (b) as an integral in the vw-plane. (f) [2 marks]. Evaluate the integral you found in part (e). [Note that your answer should agree with the one you got in part (b).]