AreaParabolicCylinderSphere::usage="The surface area between the sphere (x - 5
2
)
+
2
y
+
2
z
= 25 and the parabolic cylinder
2
z
= 2x,​ it will be obtained. Reference: Personal.";
In[]:=
Show[{​​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],ParametricPlot3D[{5Cos[u]Sin[v]+5,5Sin[u]Sin[v],5Cos[v]},{u,0,2π},{v,0,π},PlotStyle->Directive[Opacity[0.7],Specularity[White,30]],MeshFalse,ColorFunction->"BlueGreenYellow",PerformanceGoal->"Quality",PlotPoints->40]},​​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:
Surface area corresponding to the parabolic cylinder:
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+
2
z
==0;​​
2
x
+
2
y
-8x==0;​​
2
r
-8rCos[θ]==0;​​Solve[
2
r
-8rCos[θ]==0,r]​​
{{r0},{r8Cos[θ]}};​​-
π
2
<=θ<=
π
2
;
In[]:=
A
1
=2(2)
π
2
∫
0
8Cos[θ]
∫
0
1+
1
2rCos[θ]
rrθ
1
4
(60+289ArcTan[4]);
Surface area corresponding to the sphere:
In[]:=
g[x_,y_]:=
25-
2
(x-5)
-
2
y
;​​
g
x
[x_,y_]=D[g[x,y],x];​​
g
y
[x_,y_]=D[g[x,y],y];​​
1+
2
g
x
[x,y]
+
2
g
y
[x,y]
//FullSimplify
5
-
1
(-10+x)x+
2
y
;
In[]:=
A
2
=2(2)(5)
π
2
∫
0
8Cos[θ]
∫
0
-
1
2
r
-10rCos[θ]
rrθ
20(-4+10ArcTan[2]);
Total area:
A
T
=
1
4
(60+289ArcTan[4])+20(-4+10ArcTan[2]);​​
A
T
=
1
4
(-260+800ArcTan[2]+289ArcTan[4]);​​
Method 2:
In[]:=
ℜ=ImplicitRegion[
2
(x-5)
+
2
y
+
2
z
<=25&&
2
z
>=2x,{x,y,z}];​​SurfaceArea[ℜ]
1
4
(-260+800ArcTan[2]+289ArcTan[4]);
Graph:
In[]:=
Solve[{
2
(x-5)
+
2
y
+
2
z
==25,
2
z
==2x},{x,y}]
x
2
z
2
,y-
1
2
16
2
z
-
4
z
,x
2
z
2
,y
1
2
16
2
z
-
4
z
;
In[]:=
Reduce[16
2
z
-
4
z
>=0,z]
-4≤z≤4;
In[]:=
gr=Showν[t_]:=
2
t
2
,-
1
2
16
2
t
-
4
t
,t;
ν
1
[t_]:=
2
t
2
,
1
2
16
2
t
-
4
t
,t;​​{RegionPlot3D[
2
(x-5)
+
2
y
+
2
z
<=25&&
2
z
>=2x,{x,0,8},{y,-4,4},{z,-5,5},Mesh->None,PlotStyle->Directive[Green,Opacity[0.3],Specularity[White,40]],PerformanceGoal->"Quality",PlotPoints->120],ParametricPlot3D[{ν[t],
ν
1
[t]},{t,-4,4},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,-1},ImageSize700,PlotRange{{0,8},{-4,4},{-5,5}}
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