You have 2 hours 30 minutes to complete this exam. You are free to use results from the lectures, but you should clearly state any theorems you use. The exam is printed on both sides of the paper. Good luck! (1) let d be the region in r2 de ned by 1 x2 + y2 4 and y 0. (a) calculate. Zzd x2da by using polar coordinates. (recall that cos2( ) = 1. In polar coordinates, the given region is described by 1 r 2 and 0 . The integral of cos(2 ) from 0 to is 0, so we get (b) calculate the line integral. + x2y)dx + x3dy where d has the anticlockwise orientation. This is twice the answer to part (a), which is 15 . 1 (2) (a) find and classify the critical points of the function f (x, y) = 2y2 + 3xy 2x2. The critical points are those where f = 0.
