Suppose a region in the first octant is bounded by the coordinate planes and the plane $\displaystyle x+y+z=2$

and the density of the solid is $\displaystyle \delta(x,y,z) = 2xy$

I am asked to find center of mass.

I know how to find center of mass and moment about planes.

But how is this solid symmetric about xy and xz plane and not about yz plane? cause book says moment about $\displaystyle xy$

and $\displaystyle xz$ are equal by symmetry. ($\displaystyle M_{xy}$ and $\displaystyle M_{xz}$ are equal because of symmetry)

Is it possible to explain a bit?

You will find Plotting graph of $\displaystyle x+y+z=2, x=0, y=0$ and $\displaystyle z=0$ below.