How do you calculate the flux of the vector field $\displaystyle F=xi+yj+zk$ across a disk $\displaystyle x^2+z^2\leq 1$ in the plane $\displaystyle y=3$ with the normal vector pointing in the $\displaystyle y$ direction
The unit normal vector to the disk is $\displaystyle \vec{n} = \hat{j}$
So by definition
$\displaystyle \iint \vec{F}\cdot d\vec{S}=\iint \vec{F} \cdot \vec{n}dS$
Where $\displaystyle dS$ is the projection of the disk into the $\displaystyle x-z$ plane
$\displaystyle \iint_DydA$ but on this disk y is constant and $\displaystyle y=3$
$\displaystyle 3\iint_DdA$ since the integrand is constant the value of the integral is
$\displaystyle 3\pi$