Let $D$ be a region given as the set of $(x, y)$ with $a \leq x \leq b\quad\text{and}\quad-\Phi(x) \leq y \leq \Phi(x)$ where $Φ$ is a nonnegative continuous function on the interval $[a, b]$.
Let $f(x, y)$ be a function on $D$ such that $f(x, y) = - f(x, - y)$ for all $(x , y) \in D$.
Argue that $\displaystyle \iint_D f(x, y) dA = 0.$