Calculating the double integral with an integration area given by a hexagon

Question

The task is to calculate the simple double integral $$\iint_D xy dxdy$$ But the tricky thing is that the area of integration is given by a hexagon with all its corners on the circle $\{x^2+y^2 = 1\}$ and with a corner in the point (1,0). And this is were I can't seem to get anywere.

Answer 1

The integral is $0$ by two reasons each of which would suffice alone.

1) the integrand is odd in $x $ and the domain is symmetric wrt to $x=0$.

2) the integrand is odd in $y $ and the domain is symmetric wrt to $y=0$.