Two "theorical" questions about Line Integrals

While the majority of exercises related to Line Integrals are more enfocated to calculous and the difficulty of finding a correct parametrization or so, i have found these two questions (which may be related as they are two sections of the same exercise) which seem to be more theorical:

1. Let $C$ a closed, simple, piece-regular and positive orientated curve in $\mathbb{R}^2$. Verify $$\int_{C} -y^3 dx + (x^3 +2x+y)dy >0$$
2. Let $\mathbf{f}=(f_1,f_2):\mathbb{R}^2 \to \mathbb{R}^2$ of class $\mathbb{C}^2$, $C$ a closed, simple, piece-regular and positive orientated curve in $\mathbb{R}^2$ and $D$ the region of plane delimited by $C$. Verify $$\int_D det(J_{\mathbf{f}}(x,y))dxdy=\int_C f_1 \nabla f_2 d\mathbf{r}$$ where $J_{\mathbf{f}}$ is the Jacobian matrix of $\mathbf{f}$.

For the first one I have no clue on how to proceed as I do not see how to use the information about $C$ to this concrete case. The second one seems to be related to the Riemann-Green Theorem but I still don't get how to proceed.Thanks in advance.