Center of mass of 2D region

Since it is a thin plate and so essentially 2D, you can use the density as the third dimension.

reg = ImplicitRegion[{5 <= x <= 8, 0 <= y <= 20/x^2, 0 <= z <= 2 x^2}, {x, y, z}];
RegionCentroid[reg]//Most

out = {13/2, 1/4}

You can use region integrals:

reg=ImplicitRegion[{5<=x<=8,0<=y<=20/x^2},{x,y}];

δ[x_,y_]:=2x^2

Integrate[{x,y} δ[x,y], {x,y} ∈ reg] / Integrate[δ[x,y], {x,y} ∈ reg]

{13/2, 1/4}