AreaCylinderPlanes::usage="Determine the surface area of the solid that is formed between the cylinder ​
2
x
+
2
y
= 4 and the planes z = y and z = 2 y. Of the two resulting portions we want the one above XY plane.References: Internet: Ronny on line; Personal.";
In[]:=
ContourPlot3D[{
2
x
+
2
y
==4,z==y,z==2y},{x,-2,2},{y,-2,2},{z,-4,4}]
Development for surface area:
Method 1:
Surface area of the half of the first ellipse (
1
2
π a b):
In[]:=
Solve[{
2
x
+
2
y
==4,z==y},{x,y}]
In[]:=
x-
4-
2
z
,yz,x
4-
2
z
,yz;​​Reduce[4-
2
z
>=0,z]
-2≤z≤2;
b=2;​​a=Norm[{0,2,2}-{0,0,0}]
2
2
;
A
1
=
1
2
(2)2
2
π=2
2
π;
Surface area of the half of the second ellipse (
1
2
π a b):
In[]:=
Solve[{
2
x
+
2
y
==4,z==2y},{x,y}]
In[]:=
x-
1
2
16-
2
z
,y
z
2
,x
16-
2
z
2
,y
z
2
;​​Reduce[16-
2
z
>=0,z]
-4≤z≤4;
In[]:=
b=2;​​a=Norm[{0,2,4}-{0,0,0}]
2
5
;
A
2
=
1
2
(2)2
5
π=2
5
π;
Surface area of the cylinder portion above the XY plane:
f[x_,z_]:=
4-
2
x
;​​
f
x
[x_,z_]=D[f[x,z],x];​​
f
y
[x_,z_]=D[f[x,z],y];
y=z;
2
x
+
2
z
=4;z=
4-
2
x
;​​y=
z
2
;
2
x
+
2
z
4
=4;z=2
4-
2
x
;​​
4-
2
x
<=z<=2
4-
2
x
;​​-2<=x<=2;
In[]:=
f[x_,z_]:=
4-
2
x
;​​
f
x
[x_,z_]=D[f[x,z],x];​​
f
y
[x_,z_]=D[f[x,z],y];​​
A
3
=2
2
∫
0
2
4-
2
x
∫
4-
2
x
1+
2
f
x
[x,z]
+
2
f
y
[x,z]
zx
8;
Total surface area:
2
2
π+2
5
π+8;​​24+
2
π+
5
π;
Method 2:
In[]:=
ℜ=ImplicitRegion[
2
x
+
2
y
<=4&&z>=y&&z<=2y,{x,y,z}];​​SurfaceArea[ℜ]
24+
2
π+
5
π;
Graph:
In[]:=
gr=ShowRegionPlot3D[
2
x
+
2
y
<=4&&z>=y&&z<=2y,{x,-2,2},{y,-2,2},{z,0,4},Mesh->None,PlotStyle->Directive[Orange,Opacity[0.2],Specularity[White,40]],PlotPoints->120,PerformanceGoal->"Quality"],ParametricPlot3D-
4-
2
z
,z,z,
4-
2
z
,z,z,{z,-2,2},PlotStyleBlack/.Line[pts_,rest___]:>Tube[pts,0.013,rest],ParametricPlot3D-
1
2
16-
2
z
,
z
2
,z,
1
2
16-
2
z
,
z
2
,z,{z,-4,4},PlotStyleBlack/.Line[pts_,rest___]:>Tube[pts,0.013,rest],Graphics3D[{Black,Line[{{-2,0,0},{2,0,0}}]}]/.Line[pts_,rest___]:>Tube[pts,0.013,rest],AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),Axes->True,AxesOrigin{0,0,0},Boxed->False,BoxRatios->Automatic,ViewPoint->{4,2.5,3.},ImageSize700,PlotRange{{-2,2},{-2,2},{0,4}}
Out[]=
In[]:=
ResourceFunction["SpinShow"][gr,30]
Out[]=

Another Approach

Development for surface area:
Method 1:
Surface area of the cylinder portion above the XY plane:
Total surface area:
Method 2:
Graph: