Clear [f,g,x,y,z]
f[x_,y_]:=E^(-x^2)*Cos[x^2+y^2]
g[x_,y_]:=(2-x^2-y^2)
N[Integrate[g[x,y]-f[x,y],{x,-1,1},{y,-1,1}],4]
Plot3D[{f[x,y],g[x,y]},{x,-1,1},{y,-1,1}]