\frac{
\displaystyle \int_{0}^{2\pi} d\theta \int_{0}^{\frac{\pi}{2}} \pi\,d\phi \int_{0}^{1} \frac{39}{2} r^{2} \sin\phi \,dr \;\times\; \int_{(0,0)}^{(\sqrt{2},\,\sqrt{2})} \bigl(2 x y^{2}\,dx + 2 x^{2}\,dy\bigr)
}{
\displaystyle \int_{-2}^{2} \Bigl(x^{3} \cos\frac{x}{2} + \frac{1}{20}\Bigr)\sqrt{4 - x^{2}} \,dx \;\times\; \int_{0}^{\pi} d\theta \int_{0}^{1} 2 \rho \,d\rho
}

\frac{
\int_0^{2\p...