The most natural direction associated with a surface is the direction perpendicular to it. Therefore, the natural integral to compute over a surface adds up the normal component of $\EE$. This is called the flux of $\EE$ through the given surface (in the given direction): \begin{eqnarray*} \hbox{flux} = \int \EE \cdot \nn \, dA = \int \EE \cdot d\AA \end{eqnarray*} Be warned that some authors use two integral signs rather than one, and other letters, such as $S$ or $\sigma$, are often used in place of $A$.