In[]:=
AreaSpheresPlane::usage="Calculate the area in red (visualize the figure in 3D).​
​References: Internet: Ronny on line (figure)​; Personal.";
Development for surface area:
In[]:=
ℜ=ImplicitRegion[
2
x
+
2
y
+
2
z
<=16
2
π
&&
2
(x-2π)
+
2
y
+
2
z
>=4
2
π
&&x>=2π&&z>=0,{x,y,z}];​​SurfaceArea[ℜ]
Out[]=
-
2
3
(6
3
2
π
-29
3
π
)
In[]:=
-
2
3
6
3
2
π
-29
3
π
;
Animation:
In[]:=
Manipulate[Show[ContourPlot3D[{
2
x
+
2
y
+
2
z
==16
2
π
,
2
(x-2π)
+
2
y
+
2
z
==4
2
π
},{x,2π,4π},y,-2
3
π,2
3
π,{z,0,t1},ContourStyle->Directive[Red,Opacity[0.5],Specularity[White,40]],Mesh->None],RegionPlot3D[
2
x
+
2
y
+
2
z
<=16
2
π
&&
2
(x-2π)
+
2
y
+
2
z
>=4
2
π
&&x==2π&&z>=0,{x,2π,4π},y,-2
3
π,2
3
π,{z,0,t1},Mesh->None,PlotStyle->Directive[Red,Opacity[0.5],Specularity[White,40]]],RegionPlot3D[
2
x
+
2
y
+
2
z
<=16
2
π
&&
2
(x-2π)
+
2
y
+
2
z
>=4
2
π
&&z==0,{x,2π,4π},y,-2
3
π,2
3
π,{z,0,t1},Mesh->None,PlotPoints60,PlotStyle->Directive[Red,Opacity[0.5],Specularity[White,40]]],AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),Axes->True,Boxed->False,BoxRatios->Automatic,ViewPoint->{-1,-2.4,1.},ImageSize700,PlotRange{2π,4π},-2
3
π,2
3
π,0,2
3
π],{{t1,0.00002,"Value (t1)"},0.00001,2Sqrt[3]π,0.00001},ControlPlacement->Top,SaveDefinitions->True]
Out[]=
​
Value (t1)
Note: Thank you very much for your contributions: Gianluca Gorni and Michael Rogers. The hole disappeared!!