I got an assignment to prove that in a straight homogeneous rod, you can always choose a coordinate system in such a way that
$\int_S x_1dx_1dx_2=0 $
$\int_S x_2dx_1dx_2=0 $
$\int_S x_1x_2 dx_1dx_2=0$
where S is the cross section of the rod
Now, intuitively, I can understand that the first two expressions basically state that if you find the mass center of the rod, then all you have to do is to put the origin of the coordinate system in the mass center, and the integrals turn out to be zero. What I'm curious about is whether or not this is a good enough of an explanation, and also, a hint on how to obtain the third equality.