Question

Use Green's Theorem to evaluate
C (F · dr)
(Check the orientation of the curve before applying the theorem.)
F(x, y) = e^(2x) + x^(2)y, e^(2y) − xy^2

C is the circle x^2 + y^2 = 4 oriented clockwise

Answer 1

Do you mean

<b>F</b>(x,y) = e^(2x) + x^(2)y<b>i</b> + e^(2y) − xy^2 <b>j</b> ?

If so, then
<b>F</b>•d<b>r</b> = (e^(2x) + x^2y) dx + (e^(2y) − xy^2) dy

That makes
My = x^2
Nx = -y^2

∫F.dr = -∫∫x^2+y^2 dy dx
Using polar coordinates, that is just

-∫∫r^2 * r dr dθ

inside the circle:
r = [0,2]
θ = [0,2pi]