Example 2

Evaluate the surface integral of (1+x) dsigma on the plane surface x^2+y^2=z, between z=0 and z=4.
So, f(x,y,z)=(1+x)
z=S(x,y)=x^2+y^2


  f[x_,y_,z_]:=(1+x)
  S[x_,y_]:=x^2+y^2


  f[x,y,S[x,y]]


  Dx=D[S[x,y],x]
  Dy=D[S[x,y],y]
  dS=Sqrt[1+Dx^2+Dy^2]


  Integrate[f[x,y,S[x,y]]*dS,{x,-2,2},
     {y,-Sqrt[4-x^2],Sqrt[4-x^2]}]