rho[x_,y_,z_] := Exp[z] - x * y^2

m := Integrate[ rho[x,y,z], {z,0,1}, {y,-z,z},{x,-z,z}]

m

Myz := Integrate[ x*rho[x,y,z],  {z,0,1}, {y,-z,z},{x,-z,z}]

Mxy := Integrate[ z*rho[x,y,z],  {z,0,1}, {y,-z,z},{x,-z,z}]

Mxz :=  Integrate[ y*rho[x,y,z],  {z,0,1}, {y,-z,z},{x,-z,z}]

Myz

Mxy

Mxz

xBar := Myz / m
yBar := Mxz / m
zBar := Mxy / m

N[{xBar, yBar,zBar}]