How can I show (without calculating the triple integral) that the $y$ centroid of a $\mathbb R^3$ region enclosed by $z=x$, $x^2+y^2 =1$ and $x^2+y^2 =4$ equals $0$.
All I know is the plane $z=x$ holds for all values of $y$ so it should not have an impact on it. And the remaining domains are just circles on $xy$ plane centered at the origin thus centre then defines the centroid of y which equals $0$.
Is this enough to claim that the centroid is $0$? Moreover how can I express this mathematically?
It suffices to note that the region is symmetric with respect to the origin, since we have 2 ellipses centered at the origin on the plane z=x.
Mathematically you can observe that if P(x,y,z) is in the region also the symmetric point with respect to the origin Q(-x,-y-z) is in the region.