In[]:=
regCart=ImplicitRegion[2x^2+2z^2≤y≤8,{x,y,z}];Integrate[Sqrt[3]Sqrt[x^2+z^2],{x,y,z}∈regCart]
Out[]=
256π
5
3
In[]:=
regPolar=ParametricRegion[{ρCos[ϕ],y,ρSin[ϕ]},{{ρ,0,2Sqrt[2]},{ϕ,0,2π},{y,2ρ^2,8}}];Integrate[Sqrt[3]Sqrt[x^2+z^2],{x,y,z}∈regPolar]
Out[]=
256π
5
3