a = Plot3D[{x^2 + 3 y^2}, {x, -2, 2}, {y, -2, 2},
    RegionFunction ->
     Function[{x, y, z},  8 - x^2 - y^2 >=  x^2 + 3 y^2],
    BoxRatios -> Automatic , Mesh -> None, PlotStyle -> Opacity[0.8],
    PlotRange -> {{-2, 2}, {-2, 2}, {0, 8}}];
 b = Plot3D[{8 - x^2 - y^2}, {x, -2, 2}, {y, -2, 2},
    RegionFunction ->
     Function[{x, y, z}, x^2 + 3 y^2 <=  8 - x^2 - y^2],
    BoxRatios -> Automatic , Mesh -> None, PlotStyle -> Opacity[0.8],
   PlotRange -> {{-2, 2}, {-2, 2}, {0, 8}}];
c = Plot3D[{0}, {x, -2, 2}, {y, -2, 2},
   RegionFunction -> Function[{x, y, z}, x^2 + 2 y^2 <=  4],
   BoxRatios -> Automatic , Mesh -> None, PlotStyle -> Opacity[0.8],
   PlotRange -> {{-2, 2}, {-2, 2}, {0, 8}}];
Show[a, b, c]