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

∫∫R 2ydxdy.

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(1 + v²)).
(c) [2 marks]. Calculate the Jacobian determinant for this change of coordinates; recall this is given by:
∂(x, y)/∂(v,w) = det (∂x/∂u ∂x/∂w)
∂y/dv ∂y/∂w
(d) [2 marks]. Show the region R of the xy-plane corresponds to the region S of the vw-plane, where
S = [0,2] × [0,3].
(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).