I stumbled upon an example on surface integrals that uses symmetry, and I feel like I'm missing something that should be simple. We're dealing with a sphere $\Sigma_R$ centered on the origin with radius R. Halfway through the example, we're met with what is described as simple symmetry reasoning. (I'm sure it is, I just wish they'd described it better.)
$$ \iint_{\Sigma_R} x_1x_2 dS = \iint_{\Sigma_R} x_2 x_3 dS = 0 $$
Where $x_1$, $x_2$, and $x_3$ are the input variables, and S is the surface segment of the sphere. What I'm not sure about is how we can know that these integrals are symmetric. I understand why a symmetric integral would be zero, but as far as I know, symmetry happens with odd functions, whereas these are (as far as I can tell, even).
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