5aabc0…: “Close to a maroon butterfly”

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Let $R = \{(x,y) = -\pi \le x\le \pi , -\pi \le y\le \pi \}$ . It is perhaps surprising that $\int _{-\pi }^{\pi } \sin ^4 x \d x = \frac {3\pi }{4}$ .

Let $F : \R ^2 \to \R$ be the function $F(x,y) = \sin ^4 x \sin ^4 y$ .

Evaluate $\iint _R F \d A$ . $\iint _R F \d A = \answer {\frac {9 \pi ^2}{16}}.$