设f连续,S是曲面z=x^2+y^2含在柱体x^2+y^2

z=x^2+y^2为选择抛物面
S曲面在z=0平面的投影为平面区域D(如下图阴影部分)
则积分区域D换为极坐标为：
x=rcosθ,y=rsinθ,z=r^2；0≤r≤1,π/4≤θ≤5π/4
∴∫∫S [y^2+xyf(z)]dS
=∫∫D [y^2+xyf(z)]dxdy
=∫∫D [(rsinθ)^2+rcosθ*rsinθf(r^2)]*rdrdθ
=∫<0,1>r^3dr*∫<π/4,5π/4> (sinθ)^2dθ+∫<0,1>r^3f(r^2)dr*∫<π/4,5π/4>sinθcosθdθ
=[<0,1>r^4/4]*[<π/4,5π/4> θ/2-(sin2θ)/4]+∫<0,1>r^3f(r^2)dr*[<π/4,5π/4>(sinθ)^2/2]
=π/8+0*∫<0,1>r^3f(r^2)dr
=π/8