sup = ContourPlot3D[(x^2)+(y^2)+(z^2) == 1,{x,-0.5,0.5},{y,-0.5,0.5},{z,0,1}];
inf =Plot3D[(Sqrt[(x^2)+(y^2)]),{x,-0.5,0.5},{y,-0.5,0.5}];

Show[sup,inf]

sup2 = ContourPlot3D[(x^2)+(y^2)+(z^2) == 1,{x,-(Sqrt[2])/2,(Sqrt[2])/2},{y,-(Sqrt[2])/2,(Sqrt[2])/2},{z,0,1},Mesh->False,BoxRatios->Automatic];

inf2 =Plot3D[(Sqrt[(x^2)+(y^2)]),{x,-(Sqrt[2])/2,(Sqrt[2])/2},{y,-(Sqrt[2])/2,(Sqrt[2])/2},Mesh->False];

Show[sup2,inf2]

sup3= ContourPlot3D[(x^2)+(y^2)+(z^2) == 1,{x,-(Sqrt[2])/2,(Sqrt[2])/2},{y,-(Sqrt[2])/2,(Sqrt[2])/2},{z,0,1},Mesh->False,BoxRatios->Automatic,RegionFunction->Function[{x,y,z},(x^2)+(y^2)≤1/2],ContourStyle->{LightGray,Opacity[0.8],Specularity[White,200]},BoundaryStyle->{Gray,Thick}];

inf3 =Plot3D[(Sqrt[(x^2)+(y^2)]),{x,-(Sqrt[2])/2,(Sqrt[2])/2},{y,-(Sqrt[2])/2,(Sqrt[2])/2},Mesh->False,RegionFunction->Function[{x,y,z},(x^2)+(y^2)≤1/2],PlotStyle->{LightGray,Opacity[0.8],Specularity[White,200]},BoundaryStyle->{Gray,Thick}];

Show[sup3,inf3]

sup4= ContourPlot3D[(x^2)+(y^2)+(z^2) == 1,{x,-(Sqrt[2])/2,(Sqrt[2])/2},{y,-(Sqrt[2])/2,(Sqrt[2])/2},{z,0,1},Mesh->False,BoxRatios->Automatic,RegionFunction->Function[{x,y,z},(x^2)+(y^2)≤1/2],ContourStyle->{LightGray,Opacity[0.8],Specularity[White,200]},BoundaryStyle->{Gray,Thick}];

inf4 =Plot3D[{0,(Sqrt[(x^2)+(y^2)])},{x,-(Sqrt[2])/2,(Sqrt[2])/2},{y,-(Sqrt[2])/2,(Sqrt[2])/2},Mesh->False,RegionFunction->Function[{x,y,z},(x^2)+(y^2)≤1/2],PlotStyle->{LightGray,Opacity[0.8],Specularity[White,200]},BoundaryStyle->{Gray,Thick}];

Show[sup4,inf4]

AbsoluteTiming[Integrate[Integrate[((Sqrt[(a^2)-(r^2)])-(Sqrt[(r^2)]))r,{θ,0,2π}],{r,0,a/(Sqrt[2])}]]

AbsoluteTiming[Integrate[Integrate[(Sqrt[1-(x^2)-(y^2)])-(Sqrt[(x^2)+(y^2)]),{y,-(Sqrt[(1/2)-(x^2)]),(Sqrt[(1/2)-(x^2)])}],{x,-1/(Sqrt[2]),1/(Sqrt[2])}]]

AbsoluteTiming[Integrate[Integrate[(Sqrt[(a^2)-(x^2)-(y^2)])-(Sqrt[(x^2)+(y^2)]),{y,-(Sqrt[((a^2)/2)-(x^2)]),(Sqrt[((a^2)/2)-(x^2)])},Assumptions->a>0],{x,-a/(Sqrt[2]),a/(Sqrt[2])},Assumptions->a>0]]

γ[r_,λ_,ϕ_]={r Cos[λ]Cos[ϕ],r Cos[λ]Sin[ϕ],r Sin[λ]};

sup5=ParametricPlot3D[γ[1,λ,ϕ],{λ,π/4,π/2},{ϕ,0,2π}];

inf5=ParametricPlot3D[γ[r,π/4,ϕ],{r,0,1},{ϕ,0,2π}];

Show[sup5,inf5]

Integrate[Integrate[Integrate[1 (r^2)Cos[λ],{ϕ,0,2π}],{λ,π/4,π/2}],{r,0,a}]