How to prove $\int_{\Omega} \mathbf{y} \ dV = \mathbf{0}$ for a symmetric domain $\Omega$

Question

How can I show that this equation holds for a symmetric domain $\Omega$ \begin{equation} \int_{\Omega} \mathbf{y} \ dV = 0 \end{equation}

Where $\mathbf{y}$ is an arbitrary vector within the domain.

Answer 1

Let us suppose that $Ω=Ω_1\cup (-Ω_1)$, where $-Ω_1:=\{-x| x\in Ω_1\}$. Then we get $\int_Ω ydV=\int_{Ω_1} ydV+\int_{-Ω_1}ydV=\int_{Ω_1} ydV+\int_{Ω_1}-y(1)dV=0$, where we changed the variables from $y$ to $-y$ via transformation $T(y)=-y$ with absolute value of Jacobian $|J(T)|=1$.