Q:

Calculate double integral $\iint_A \sin (x+y) dxdy$

Calculate double integral $$\iint_A \sin (x+y) dxdy$$
where:
$$A=\{ \left(x,y \right)\in \mathbb{R}^2: 0 \le x \le \pi, 0 \le y \le \pi\}$$
How to calculate that? $x+y$ in sin is confusing as i do not know how to change it to iterated integral or maybe is there a different method to calculate that?

A:

It should be fairly easy to follow the path outlined by John Hughes' comment, but even quicker to see that the integral must be $0$ due to symmetry.
Along the diagonal $x+y=\pi$, we have $\sin(x+y)=0$. A point above the diagonal has a negative value that corresponds exactly to the positive value at the point you get by reflecting across the diagonal. (And the domain of integration $A$ is invariant under this reflection).

