Gauss' law is a very powerful method to determine the electric field due to a distribution of charges. The mathematical expression for Gauss' law is:

\[ \int_{S} \vec{E} \cdot \vec{dA}=\frac{Q_{enc}}{\epsilon_0} \]

where \(S\) is a surface, \(\vec{E}\) is the electric field vector, \(\vec{dA}\) is the infinitesimal area element, \(Q_{enc}\) is the charge enclosed by \(S\) and \(\epsilon_0\) is a constant.

We now turn to the notion of integration over a specified surface. One can think about definite integration over a line interval, i.e. \( \int_a^b f(x) dx\), as the area under the curve defined by \(f(x)\). Similarly, the integral over a surface of a function can be thought of as the volume under the 2-d graph of the function. Using this definition, what is the integral

\[\int_S x dA\]

where the surface \(S\) is the square in the \(xy\)-plane with corners at \((0,0), (1,0),(0,1),(1,1)\).

This problem is part of David's set on Gauss' Law.

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