Integration over surfaces

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:

SEdA=Qencϵ0 \int_{S} \vec{E} \cdot \vec{dA}=\frac{Q_{enc}}{\epsilon_0}

where SS is a surface, E\vec{E} is the electric field vector, dA\vec{dA} is the infinitesimal area element, QencQ_{enc} is the charge enclosed by SS and ϵ0\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. abf(x)dx \int_a^b f(x) dx, as the area under the curve defined by f(x)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

SxdA\int_S x dA

where the surface SS is the square in the xyxy-plane with corners at (0,0),(1,0),(0,1),(1,1)(0,0), (1,0),(0,1),(1,1).


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

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