Clear[x,y]
f[x_,y_]=(E^(-(x^2)))*Cos[(x^2)+(y^2)]
g[x_,y_]=2-(x^2)-(y^2)
Plot3D[{f[x,y],g[x,y]},{x,-1,1},{y,-1,1}]
N[(∫_(-1)^1)((∫_(-1)^1)f[x,y]\[DifferentialD]x)\[DifferentialD]y-(∫_(-1)^1)((∫_(-1)^1)g[x,y]\[DifferentialD]x)\[DifferentialD]y]