# Interesting question about line integrals

Let $a_1,a_2,a_3,a_4 \in \mathbb{R}$ be given so that $(a_1,a_2)$ and $(a_3,a_4)$ are linearly independent vectors in the plane. Let $\alpha$ be the boundary of the parallelogram with vertices $(0,0), (a,b),(c,d),(a+c,b+d)$ oriented counterclockwise. How can I obtain

$$\int_{\alpha} x dy - y dx$$

Green's theorem says $\int_C Ldx+Mdy=\int\int_D\frac{\partial M}{\partial x}-\frac{\partial L}{\partial y}dA$, where C is a closed loop bounding a region without singularities.
In this case $\int_C -ydx+xdy=\int\int_D\frac{\partial x}{\partial x}-\frac{\partial (-y)}{\partial y}dA=\int\int_D2dxdy$
$D$ is the region bounded by the parallelogram, you can calculate the integral by Fubini's Theorem. Since it's a parallelogram with opposite sides parallel, change the coordinate to make the integration to be over a rectangular region is even better.