h[x_,y_]=4-(x^2)-(y^2);

Plot3D[{h[x,y],0},{x,-3,3},{y,-3,3},RegionFunction->Function[{x,y,z},h[x,y]≥0]]

α1[r_,θ_]={r Cos[θ],r Sin[θ],4-(r^2)};

α2[r_,θ_]={r Cos[θ],r Sin[θ],0};

ParametricPlot3D[{α1[r,θ],α2[r,θ]},{r,0,2},{θ,0,2 π}]

w[x_,y_,z_]={x,y,1+(z^2)}

divw[x_,y_,z_]=D[w[x,y,z][[1]],x]+D[w[x,y,z][[2]],y]+D[w[x,y,z][[3]],z]

Integrate[divw[x,y,z],{z,0,4-(x^2)-(y^2)}]

Factor[%]

%/.((x^2)+(y^2))->(r^2)

Integrate[%*r,{r,0,2}]

Integrate[%,{θ,0,2 π}]

vmer2=D[α2[r,θ],r]

vpar2=D[α2[r,θ],θ]

ds2[r_,θ_]=Simplify[Cross[vmer2,vpar2]]

Simplify[w[α2[r,θ][[1]],α2[r,θ][[2]],α2[r,θ][[3]]].ds2[r,θ]]

Integrate[%,{r,0,2}]

Integrate[%,{θ,0,2 π}]

vmer1=D[α1[r,θ],r]

vpar1=D[α1[r,θ],θ]

ds1[r_,θ_]=Simplify[Cross[vmer1,vpar1]]

{2 (r^2) Cos[θ],2 (r^2) Sin[θ],r}
w[α1[r,θ][[1]],α1[r,θ][[2]],α1[r,θ][[3]]]

Simplify[w[α1[r,θ][[1]],α1[r,θ][[2]],α1[r,θ][[3]]].ds1[r,θ]]

Integrate[%,{r,0,2}]

Integrate[%,{θ,0,2 π}]

%-4 π

Show[Graphics3D[Arrow[{{0,0,4},{0,0,5}}]],Graphics3D[Arrow[{{0,1,0},{0,1,-1}}]],ParametricPlot3D[{α1[r,θ],α2[r,θ]},{r,0,2},{θ,0,2 π},PlotStyle->Opacity[0.2]]]

ParametricPlot3D[α2[r,θ],{r,0,2},{θ,0,2π},PlotStyle->Opacity[0.5],Boxed->False]

a=ParametricPlot3D[α2[r,θ],{r,0,2},{θ,0,2π},PlotStyle->Opacity[0.5],Mesh->None,Boxed->False];

b=ParametricPlot3D[{α2[r,0],α2[r,π/8]},{r,0,2},Boxed->False];

c=ParametricPlot3D[{α2[1,θ],α2[1.5,θ]},{θ,0,2π},Boxed->False];

Show[a,b,c,Graphics3D[Arrow[{α2[1,0],α2[1.5,0]}]],Graphics3D[Arrow[{α2[1,0],α2[1,π/8]}]],Graphics3D[Arrow[{α2[1,0],α2[1,0]+{0,0,0.6}}]]]

ParametricPlot3D[α1[r,θ],{r,0,2},{θ,0,2π},PlotStyle->Opacity[0.5],Boxed->False]

2π (17(((2))^2)/2-6(((2))^4)/4 + (2^6)/6)

%-4π