AreaParabolicCylinderCylinderPlaneXY::usage="The surface area between the cylinder (x-5
2
)
+
2
y
= 25 and the parabolic cylinder
2
z
= 2x,​ above the XY plane, it will be obtained. Reference: Personal.";
In[]:=
Show[{​​ParametricPlot3D[{5Cos[t]+5,5Sin[t],v},{t,0,2π},{v,-9,9},PlotStyle->Directive[Opacity[0.7],Specularity[White,30]],MeshFalse,ColorFunction->"BlueGreenYellow",PerformanceGoal->"Quality",PlotPoints->40],​​ContourPlot3D[
2
z
==2x,{x,0,10},{y,-15,15},{z,-6,6},ContourStyle->Directive[Opacity[0.2],Specularity[White,30]],MeshFalse,ColorFunction->"BlueGreenYellow",PerformanceGoal->"Quality",PlotPoints->40],ContourPlot3D[z==0,{x,-10,10},{y,-10,10},{z,-5,5},MeshNone,ContourStyle->{Green,Opacity[0.8]}]},​​AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,ImageSize900,ViewPoint{-1.3,-2.4,1}]
Out[]=
Development for surface area:
Method 1:
Viviani’s vault:
In[]:=
​​f[x_,y_]:=
2x
;​​
f
x
[x_,y_]=D[f[x,y],x];​​
f
y
[x_,y_]=D[f[x,y],y];​​
1+
2
f
x
[x,y]
+
2
f
y
[x,y]
//FullSimplify
1+
1
2x
;
In[]:=
x=rCos[θ];y=rSin[θ];​​
2
x
+
2
y
-10x==0;​​
2
r
2
Cos[θ]
+
2
r
2
Sin[θ]
-10rCos[θ]==0;​​r(r-10Cos[θ])==0;​​0<=r<=10Cos[θ];​​-
π
2
<=θ<=
π
2
;​​
A
1
=2
π
2
∫
0
10Cos[θ]
∫
0
1+
1
2rCos[θ]
rrθ
1
8
38
5
+441ArcTan[2
5
];
Cylinder base:
A
2
=π
2
(5)
=25π;
Portion of the cylinder:
In[]:=
Solve[{
2
(x-5)
+
2
y
==25,
2
z
==2x},{y,z}]
y-
10x-
2
x
,z-
2
x
,y-
10x-
2
x
,z
2
x
,y
10x-
2
x
,z-
2
x
,y
10x-
2
x
,z
2
x
;
In[]:=
g[x_,z_]:=
25-
2
(x-5)
;​​
g
x
[x_,z_]=D[g[x,z],x];​​
g
z
[x_,z_]=D[g[x,z],z];​​
1+
2
g
x
[x,z]
+
2
g
z
[x,z]
//FullSimplify​​
5
1
10x-
2
x
;
By symmetry:
In[]:=
A
3
=2(5)
10
∫
0
2
x
∫
0
1
10x-
2
x
zx
40
5
;
Total area:
A
T
=
1
8
38
5
+441ArcTan[2
5
]+25π+40
5
;
1
8
358
5
+200π+441ArcTan[2
5
];
Method 2:
In[]:=
ℜ=ImplicitRegion[
2
(x-5)
+
2
y
<=25&&
2
z
<=2x&&z>=0,{x,y,z}];​​SurfaceArea[ℜ]
1
8
358
5
+200π+441ArcTan[2
5
];
Graph:
In[]:=
Solve[{
2
(x-5)
+
2
y
==25,
2
z
==2x},{x,y}]
x
2
z
2
,y-
1
2
20
2
z
-
4
z
,x
2
z
2
,y
1
2
20
2
z
-
4
z
;
In[]:=
Reduce[20
2
z
-
4
z
>=0,z]
-2
5
≤z≤2
5
;
Only for z ≥ 0:
0≤z≤2
5
;
In[]:=
gr=Showν[t_]:=
2
t
2
,-
1
2
20
2
t
-
4
t
,t;
ν
1
[t_]:=
2
t
2
,
1
2
20
2
t
-
4
t
,t;​​RegionPlot3D[
2
(x-5)
+
2
y
<=25&&
2
z
<=2x&&z>=0,{x,0,10},{y,-5,5},z,0,2
5
,Mesh->None,PlotStyle->Directive[Green,Opacity[0.3],Specularity[White,40]],PerformanceGoal->"Quality",PlotPoints120,BoundaryStyleDirective[Black,Thick]],ParametricPlot3D[{ν[t],
ν
1
[t]},t,0,2
5
,PlotStyleBlack]/.Line[pts_,rest___]:>Tube[pts,0.026,rest],AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),Axes->True,AxesOrigin{0,0,0},Boxed->False,BoxRatios->Automatic,ViewPoint->{1.3,-2.4,2.},ImageSize700,PlotRange{0,10},{-5,5},0,2
5

Out[]=