Example 2.3.10: Flux through a rectangle in the yz-plane

The surface is the rectangle of Example 2.3.4. Its orientation is $a_x$, so $dS=a_{x}dydz$ and only the normal component $A_x=2y^{2}z$ contributes:

(On $x=0$ the tangential component $A_y=xy$ is zero anyway, and $A_z$ is tangential to the surface.) The integrand is a product of a function of $y$ and a function of $z$, and the limits are constants, so the integral factorises:

Check: on the rectangle $A_x=2y^{2}z\ge 0$, so the flux in the direction $a_x$ must be positive, and it must vanish when either side $a$ or $b$ shrinks to zero. Both hold. With the opposite orientation, $-a_x$, the flux would be $-a^3{b}^2/3$.