Suppose a double integral $$ I_{D}:=\iint\limits_D f(x,y) dx dy $$ over a specific type of region $D=\{(x,y) : x_{\min}\leq x \leq x_{\max}, h_1(x) \leq y \leq h_2(x)\}$. Note that $h_1, h_2$ are periodic continuous functions over the $x$-interval of the region $D$, i.e., $h_1(x_{\min}) = h_1(x_{\max})$. $h_2(x_{\min}) = h_2(x_{\max})$. Furthermore the functions $h_1, h_2$ never cross each other, that is, $\min\limits_{x} h_2(x) > \max\limits_{x} h_1(x)$.
For the sake of notational simplicity, let $y_{\max}:=\max\limits_{x} h_2(x)$ and $y_{\min}:=\min\limits_{x} h_1(x)$. Let $B:=\{(x,y):x_{\min}\leq x \leq x_{\max}, a< y < b\}$ denote a box that spans the same range in the $x$ axis as $D$. If the double integral $I$ over $B$ is the same for all $a,b$, can we infer anything about the sign of $D$?
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