Why does this double integral have the value zero?

Question

Why does the following double integral have the value zero (no calculation needed) $$\iint_Dx^8y^7\,dxdy=0$$ where $D={x^2+y^2 \le 2, x \ge0}$.
Does this have anything to do with symmetry, odd and even functions? Can anyone help me with how to think and understand integrals more, especially how to use symmetry to facilitate integral calculations.

Any useful links or videos would be appreciated as well-

Answer 1

Because the map $y \mapsto y^7$ is odd and the domain $D$ symmetric with regards to the $y$-axis.

In general for an odd map, $\int_{-a}^a f(x) \ dx=0$ for $a \gt 0$.

Answer 2

Because for every point (x,y) in your D, there exists another point (x,-y) for which your integral would yield the exact negative value. So everything gets canceled out.