AreaSpheresPlane::usage="Calculate the area in red (visualize the figure in 3D).​
​References: Internet: Ronny on line (figure)​; Personal.";
Development for surface area:
Method 1:
In[]:=
A1=
π
2
2
3
π
-π
2
(2π)
2
​​A2=2
4π
∫
2π

16
2
π
-
2
x
-
4
2
π
-
2
(x-2π)
x​​
4
3
π
;
2-2
3
2
π
+
5
3
π
3
;
In[]:=
f[x_]:=
16
2
π
-
2
x
;​​A3=
4π
∫
2π
πf[x]
1+
2
f'[x]
x
8
3
π
;
In[]:=
g[x_]:=
4
2
π
-
2
(x-2π)
;​​A4=
4π
∫
2π
πg[x]
1+
2
g'[x]
x
4
3
π
;
In[]:=
A=4
3
π
+2-2
3
2
π
+
5
3
π
3
+8
3
π
+4
3
π
//FullSimplify
Total area:
2
3
2
π
-6
3
+29π;
Method 2:
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]
-
2
3
6
3
2
π
-29
3
π
;​​
2
3
2
π
-6
3
+29π;
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]
Out[]=
​
Value (t1)
Note: Thank you very much for your contributions: Gianluca Gorni and Michael Rogers. The hole disappeared!!
​​​​​A new and very interesting contribution by Michael Rogers:
In[]:=
gr=Blockt1=2
3
π-0.00001,xform[pts_]:=Module{x,y,z,n},{x,y,z}=Transpose[pts];​​n=Sqrt4
2
π
-
2
Min
2π
t1
,1z
Sqrt[
2
(x-2π)
+
2
y
];​​Transpose@n(x-2π)+2π,ny,Min
2π
t1
,1z;​​plot=ParametricPlot3D4π{Sin[u]Cos[w],Sin[u]Sin[w],Cos[u]},u,ArcCos
t1
4π
,
π
2
,w,ArcCos
Csc[u]
2
,-ArcCos
Csc[u]
2
,Mesh->None,PlotStyle->{Red,Opacity[0.5],Specularity[White,40]},PlotPoints->{Ceiling[5+8t1],25},BoundaryStyle->Black,WorkingPrecision->24,Method->{"BoundaryOffset"->False};​​(*Sorttoensurevertex-to-vertexorientation(y=#[[2]],z=Last)*)bdypts=First@Cases[Normal@plot,Line[p_]:>p,Infinity];​​outerX2PiNeg=SortBy[Cases[bdypts,{x_,y_,z_}/;x==2π&&y<=0],Last];​​outerX2PiPos=SortBy[Cases[bdypts,{x_,y_,z_}/;x==2π&&y>=0],-Last[#]&];​​outerZ0=SortBy[Cases[bdypts,{x_,y_,z_}/;z+1==1.],-#[[2]]&];​​outerZT1=SortBy[Cases[bdypts,{x_,y_,z_}/;z==t1],#[[2]]&];​​{innerX2PiNeg,innerX2PiPos,innerZ0,innerZT1}=Reverse/@xform/@{outerX2PiNeg,outerX2PiPos,outerZ0,outerZT1};​​plot=plot/._Line->{};(*removeboundaryline*)Show[Graphics3D[{},PlotRange->All,Axes->True,AxesLabel->{x,y,z},AxesOrigin->{0,0,0},Boxed->False,BoxRatios->Automatic,ViewPoint->{-1,-2.4,1.},ImageSize700,PlotRange{2π,4π},-2
3
π,2
3
π,0,2
3
π],plot,plot/.GraphicsComplex[pts_,rest___]:>With[{xpts=xform[pts]},GraphicsComplex[xpts,rest]/.(VertexNormals->_)->VertexNormals->Normalize/@(#-{2π,0,0}&/@xpts)],Graphics3D[{Red,Opacity[0.5],{Polygon[Join[outerX2PiNeg,outerX2PiPos,innerX2PiPos,innerX2PiNeg]],Polygon[Join[outerZ0,innerZ0]]},If[t1<2π,Polygon[Join[outerZT1,innerZT1]],Polygon[outerZT1]]}],PlotLabel->HoldForm[t1]==t1]