Example

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

In[44]:=

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

In[45]:=

  h[x,y,f[x,y]]

Out[45]=

  1 + x

In[46]:=

  Dx=D[f[x,y],x]
  Dy=D[f[x,y],y]
  dA=Sqrt[1+Dx^2+Dy^2]

Out[46]=

2 x

Out[47]=

2 y

Out[48]=

              2      2
  Sqrt[1 + 4 x  + 4 y ]

In[49]:=

  Integrate[h[x,y,f[x,y]]*dA,{x,-2,2},
     {y,-Sqrt[4-x^2],Sqrt[4-x^2]}]

Out[49]=

  -Pi   17 Sqrt[17] Pi
  --- + --------------
   6          6