I need help with this problem. The problem is attached.
From Green's theorem
$\displaystyle
\int \limits_C -y^3\,dx +(x^3+2x+y)\,dy = \iint \limits_R 3x^2+2+3y^2\,dA
$
and since
$\displaystyle 3x^2 + 2 +3y^2 > 0$,
then
$\displaystyle \iint \limits_R \left(3x^2 + 2 +3y^2 \right) dA > 0\;\; \Rightarrow\;\; \int \limits_C -y^3\,dx +(x^3+2x+y)\,dy > 0$ .