Find circulation
$\displaystyle F(x,y) = (x+2xy)i + (y+x^2)j $ around the
$\displaystyle \alpha : |x| + |y| = 1$
Use Green's Theorem which says the circulation of a vector field around a simpled closed curve in the xy-plane is equal to the line integral of the vector field evaluated around the curve. In ths problem the curve is diamond-shaped. So you have to parametrize each side and evaluate four separate line integrals.
BTW, is the answer to that flux problem in your textbook?
Starting at (1,0) and and going counterlcockwise
$\displaystyle \int \int_{D} (\nabla \times F) \cdot \hat{k} \ dA $= $\displaystyle \int_{C} F \cdot dr = \int_{0}^{1} \Big( F(1-t,t) \cdot (-1,1) \Big) dt + \int_{0}^{1} \Big(F(-t,1-t) \cdot (-1,-1) \Big) dt $ $\displaystyle + \int_{0}^{1} \Big(F(t-1, -t) \cdot (1,-1) \Big) dt + \int_{0}^{1} \Big( F(t, t-1) \cdot (1,1) \Big) dt $