AreaParabolicCylinderCone::usage="The surface area between the cone 9
2
x
+ 9
2
y
- 29
2
z
= 0 and the parabolic cylinder
2
z
= 2x,​ it will be obtained. Reference: Personal.";
In[]:=
ShowParametricPlot3D
29
3
vCos[u],
29
3
vSin[u],v,{u,0,2π},{v,-6,6},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[0.5],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,-2.4,0.5}
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
;
x=rCos[θ];y=rSin[θ];​​9
2
x
+9
2
y
-29
2
z
==0;​​9
2
x
+9
2
y
-58x==0;​​9
2
r
-58rCos[θ]==0;​​Solve[9
2
r
-58rCos[θ]==0,r]​​
{r0},r
58Cos[θ]
9
;​​-
π
2
<=θ<=
π
2
;
A
1
=2(2)
π
2
∫
0
58Cos[θ]
9
∫
0
1+
1
2rCos[θ]
rrθ
1
324
642
29
+15625ArcTan
2
29
3
;
Surface area corresponding to the cone:
In[]:=
g[x_,y_]:=
3
29
2
x
+
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
38
29
;
In[]:=
A
2
=2(2)
38
29
π
2
∫
0
58Cos[θ]
9
∫
0
rrθ
58
1102
π
81
;
Total area:
A
T
=
1
324
642
29
+15625ArcTan
2
29
3
+
58
1102
π
81
;​​
A
T
=
1
324
642
29
+232
1102
π+15625ArcTan
2
29
3
;​​
Method 2:
In[]:=
ℜ=ImplicitRegion[9
2
x
+9
2
y
-29
2
z
<=0&&
2
z
<=2x,{x,y,z}];​​SurfaceArea[ℜ]
1
324
642
29
+232
1102
π+15625ArcTan
2
29
3
;
Graph:
In[]:=
Solve[{9
2
x
+9
2
y
-29
2
z
==0,
2
z
==2x},{x,y}]
x
2
z
2
,y-
1
6
116
2
z
-9
4
z
,x
2
z
2
,y
1
6
116
2
z
-9
4
z
;
In[]:=
Reduce[116
2
z
-9
4
z
>=0,z]
-
2
29
3
≤z≤
2
29
3
;
Showν[t_]:=
2
t
2
,-
1
6
116
2
t
-9
4
t
,t;
ν
1
[t_]:=
2
t
2
,
1
6
116
2
t
-9
4
t
,t;​​RegionPlot3D9
2
x
+9
2
y
-29
2
z
<=0&&
2
z
<=2x,x,0,
58
9
,y,-
29
9
,
29
9
,z,-
2
29
3
,
2
29
3
,Mesh->None,PlotStyle->Directive[Green,Opacity[0.3],Specularity[White,40]],PerformanceGoal->"Quality",PlotPoints->300,ParametricPlot3D{ν[t],
ν
1
[t]},t,-
2
29
3
,
2
29
3
,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,
58
9
,-
29
9
,
29
9
,-
2
29
3
,
2
29
3

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