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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 $$\displaystyle \iint_D f(x, y) dA = 0.$$

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