2. For each of the following, evaluate Ie w.

(a) C is the topological boundary of the rectangle [a, b] x [c, d], oriented in the counterclockwise direction, and w = (f(x) + y) dx + xy dy, where f : [0,1]-T R is any continuous function.

(b) C is the topological boundary of the two-dimensional region bounded by y =

x2 and y = x, oriented in the clockwise direction, and w = yf(x) dx + (x2 +

y2) dy, where f : [0,1] -T R is C1 and satisfies Io1 xf(x) dx = Io1 x2 f(x) dx.

(c) C is the topological boundary of a two-dimensional region E that satisfies the hypotheses of Green's Theorem, oriented positively, and w = eX sin y dyeX cosydx.