VolumeCylinders::usage="Calculate the volume between the two cylinders:​
2
x
+
2
z
= 9 and
2
y
+
2
z
= 9 in the first octant.References: Internet: Ronny on line; Personal.";
Development for volume:
Method 1 (See development for the graph below):
By symmetry:
In[]:=
V=2
3
∫
0
x
∫
0
9-
2
x
yx
Out[]=
18
Method 2:
In[]:=
ℜ=ImplicitRegion[
2
x
+
2
z
<=9&&
2
y
+
2
z
<=9&&y>=0&&z>=0&&x>=0,{x,y,z}];​​Volume@RegionIntersection[ℜ]
Out[]=
18
Development for the graph:
In[]:=
Solve[{
2
x
+
2
z
==9,
2
y
+
2
z
==9},{y,z}]
y-x,z-
9-
2
x
,yx,z-
9-
2
x
,y-x,z
9-
2
x
,yx,z
9-
2
x
;
In[]:=
Reduce[9-
2
x
>=0,x]
-3≤x≤3;​​0<=x<=3;
It’s only the first octant.
Animation:
In[]:=
Manipulateζ[t_]:=t,t,
9-
2
t
;
ζ
1
[t_]:=t,0,
9-
2
t
;
ζ
2
[t_]:=0,t,
9-
2
t
;Show[{RegionPlot3D[
2
x
+
2
z
<=9&&
2
y
+
2
z
<=9&&y>=0&&z>=0&&x>=0,{x,0,3},{y,0,3},{z,0,t1},Mesh->None,PerformanceGoal->"Quality",PlotPoints60,BoundaryStyleDirective[Black,Thick],PlotStyleDirective[Green,Opacity[0.3],Specularity[White,40]]],ParametricPlot3D[ζ[t],{t,0,3},PlotStyleBlack]/.Line[pts_,rest___]:>Tube[pts,0.005,rest],ParametricPlot3D[
ζ
1
[t],{t,0,3},PlotStyleBlack]/.Line[pts_,rest___]:>Tube[pts,0.005,rest],​​ParametricPlot3D[
ζ
2
[t],{t,0,3},PlotStyleBlack]/.Line[pts_,rest___]:>Tube[pts,0.005,rest],Graphics3D[{Black,Thick,Line[{{0,0,0},{0,0,3}}]}],Graphics3D[{Black,Thick,Line[{{3,0,0},{3,3,0}}]}],Graphics3D[{Black,Thick,Line[{{0,0,0},{3,3,0}}]}]},AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRange{{0,3},{0,3},{0,3}},ViewPoint{1.3,2.4,1.5},ImageSize700],Style["Volume between two cylinders:",Bold,Large],{{t1,0.00002,"Value (t1)"},0.00001,3,0.00001},ControlPlacementTop
Out[]=
​
Volume between two cylinders:
Value (t1)