AreaCylinderConePlaneXY::usage="The surface area between the cylinder (x-5
2
)
+
2
y
= 25 and the cone 9
2
x
+ 9
2
y
- 29
2
z
= 0,​ above the XY plane, it will be obtained. Reference: Personal.";
In[]:=
ShowParametricPlot3D
29
3
vCos[u],
29
3
vSin[u],v,{u,0,2π},{v,-9,9},PlotStyle->Directive[Opacity[0.7],Specularity[White,30]],MeshFalse,ColorFunction->"BlueGreenYellow",PerformanceGoal->"Quality",PlotPoints->40,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[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700,ViewPoint{1.3,-2.4,1}
Out[]=
Development for surface area:
Method 1:
Viviani’s vault:
In[]:=
​​f[x_,y_]:=
3
29
2
x
+
2
y
;​​
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
38
29
;
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
38
29
π
2
∫
0
10Cos[θ]
∫
0
rrθ
25
38
29
π;
Cylinder base:
A
2
=π
2
(5)
=25π;
Portion of the cylinder:
In[]:=
Solve[Solve[{
2
(x-5)
+
2
y
==25,9
2
x
+9
2
y
-29
2
z
==0},{y,z}]]
Solvey-
10x-
2
x
,z-3
10
29
x
,y-
10x-
2
x
,z3
10
29
x
,y
10x-
2
x
,z-3
10
29
x
,y
10x-
2
x
,z3
10
29
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
3
10
29
x
∫
0
1
10x-
2
x
zx
600
29
;
Total area:
A
T
=
600
29
+25π+25
38
29
π;
Method 2:
In[]:=
ℜ=ImplicitRegion[
2
(x-5)
+
2
y
<=25&&9
2
x
+9
2
y
-29
2
z
>=0&&z>=0,{x,y,z}];​​SurfaceArea[ℜ]
600
29
+25π+25
38
29
π;
Graph:
In[]:=
Solve[{
2
(x-5)
+
2
y
==25,9
2
x
+9
2
y
-29
2
z
==0},{x,y}]
x
29
2
z
90
,y-
1
90
29
900
2
z
-29
4
z
,x
29
2
z
90
,y
1
90
29
900
2
z
-29
4
z
;
In[]:=
Reduce[900
2
z
-29
4
z
>=0,z]
-
30
29
≤z≤
30
29
;
Only for z ≥ 0:
0≤z≤
30
29
;
In[]:=
gr=Showν[t_]:=
29
2
t
90
,-
1
90
29
900
2
t
-29
4
t
,t;
ν
1
[t_]:=
29
2
t
90
,
1
90
29
900
2
t
-29
4
t
,t;​​RegionPlot3D
2
(x-5)
+
2
y
<=25&&9
2
x
+9
2
y
-29
2
z
>=0&&z>=0,{x,0,10},{y,-5,5},z,0,
30
29
,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,
30
29
,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,
30
29

Out[]=