Flux through a spherical quadrant by planar caps ($\Phi =\pi R^4(3R^2-2)/16$)

For the outward spherical patch $r=R$, $y\le 0$, $z\le 0$, and the polynomial vector field $F=(y{x}^2,x{y}^2-3z^4,x^3+y^2)$, close the quarter-ball with its two planar half-discs. The divergence is $4xy$, whose volume integral vanishes by symmetry in $x$. The caps have fluxes $-3\pi R^6/16$ and $\pi R^4/8$, obtained from half-disc fourth and second moments. The divergence theorem gives the displayed spherical flux. All pieces use normals pointing out of the same closed volume; the caps are auxiliary and must be subtracted.