AreaConeSphere::usage="The surface area between the sphere (x - 9
2
)
+
2
y
+
2
z
= 81 and the cone 9
2
x
+ 9
2
y
- 29
2
z
= 0,​ it will be obtained. Reference: Personal.";
In[]:=
Show​​ParametricPlot3D[{9Cos[u]Sin[v]+9,9Sin[u]Sin[v],9Cos[v]},{u,0,2π},{v,0,π},PlotStyle->Directive[Opacity[0.7],Specularity[White,30]],MeshFalse,ColorFunction->"BlueGreenYellow",PerformanceGoal->"Quality",PlotPoints->40],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,​​AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,ImageSize700,ViewPoint{1.3,-5,3}
Out[]=
Development for surface area:
Method 1:
Surface area corresponding to the cone:
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
;
In[]:=
x=rCos[θ];y=rSin[θ];​​
38
29
2
x
+
38
29
2
y
-18x==0;​​38
2
x
+38
2
y
-522x==0;​​38
2
r
-522rCos[θ]==0;​​Solve[38
2
r
-522rCos[θ]==0,r]
{r0},r
261Cos[θ]
19
;​​-
π
2
<=θ<=
π
2
In[]:=
A
1
=2(2)
38
29
π
2
∫
0
261
19
Cos[θ]
∫
0
rrθ
2349
19
29
38
π;
Surface area corresponding to the sphere:
In[]:=
g[x_,y_]:=
81-
2
(x-9)
-
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
9
-
1
(-18+x)x+
2
y
;
In[]:=
Solve[{
2
(x-9)
+
2
y
+
2
z
==81,9
2
x
+9
2
y
-29
2
z
==0},{y,z}]
y-
261x-19
2
x
19
,z-
9
x
19
,y-
261x-19
2
x
19
,z
9
x
19
,y
261x-19
2
x
19
,z-
9
x
19
,y
261x-19
2
x
19
,z
9
x
19
;
In[]:=
A
2
=2(9)
261
19
∫
0
261x-19
2
x
19
∫
-
261x-19
2
x
19
-
1
(-18+x)x+
2
y
yx
18-
54
29
19
+36ArcCot
3
29
;
In[]:=
A
T
=
2349
19
29
38
π+18-
54
29
19
+36ArcCot
3
29
//FullSimplify​​
81
722
-456
29
+29
1102
π+5776ArcCot
3
29
;
Method 2:
In[]:=
ℜ=ImplicitRegion[
2
(x-9)
+
2
y
+
2
z
<=81&&9
2
x
+9
2
y
-29
2
z
<=0,{x,y,z}];​​SurfaceArea[ℜ]
81
722
-456
29
+29
1102
π+5776ArcCot
3
29
;
Graph:
In[]:=
Solve[{
2
(x-9)
+
2
y
+
2
z
==81,9
2
x
+9
2
y
-29
2
z
==0},{x,y}]
x
19
2
z
81
,y-
1
81
21141
2
z
-361
4
z
,x
19
2
z
81
,y
1
81
21141
2
z
-361
4
z
;
In[]:=
Reduce[21141
2
z
-361
4
z
>=0,z]
-
27
29
19
≤z≤
27
29
19
;
In[]:=
gr=Showν[t_]:=
19
2
t
81
,-
1
81
21141
2
t
-361
4
t
,t;
ν
1
[t_]:=
19
2
t
81
,
1
81
21141
2
t
-361
4
t
,t;​​RegionPlot3D
2
(x-9)
+
2
y
+
2
z
<=81&&9
2
x
+9
2
y
-29
2
z
<=0,x,0,
261
19
,y,-
261
38
,
261
38
,{z,-9,9},Mesh->None,PlotStyle->Directive[Green,Opacity[0.3],Specularity[White,40]],PerformanceGoal->"Quality",PlotPoints->120,ParametricPlot3D{ν[t],
ν
1
[t]},t,-
27
29
19
,
27
29
19
,PlotStyleBlack/.Line[pts_,rest___]:>Tube[pts,0.052,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,
261
19
,-
261
38
,
261
38
,{-9,9}
Out[]=