VolumeEllipticCylinderPlane::usage="Calculate the volume between the elliptic cylinder​
2
z
+ 4
2
y
= 4 and the plane x + z = 2 in the first octant.References: Internet: Ronny on line; Personal.";
Development for volume:
Method 1 (See development for the graph below):
In[]:=
V=
1
∫
0
2
1-
2
y
∫
0
2-z
∫
0
xzy
Out[]=
-
4
3
+π
Method 2:
In[]:=
ℜ=ImplicitRegion[
2
z
+4
2
y
<=4&&x+z<=2&&y>=0&&z>=0&&x>=0,{x,y,z}];​​Volume@RegionIntersection[ℜ]
Out[]=
-
4
3
+π
Development for the graph:
In[]:=
Solve[{
2
z
+4
2
y
==4,x+z==2},{x,z}]
x21+
1-
2
y
,z-2
1-
2
y
,x21-
1-
2
y
,z2
1-
2
y
;
In[]:=
Reduce[1-
2
y
>=0,y]
-1≤y≤1;​​0<=y<=1;
It’s only the first octant.
Animation:
In[]:=
Manipulateζ[t_]:=21-
1-
2
t
,t,2
1-
2
t
;
ζ
1
[t_]:=0,t,2
1-
2
t
;Show[{RegionPlot3D[
2
z
+4
2
y
<=4&&x+z<=2,{x,0,2},{y,0,1},{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,0,1},PlotStyleBlack]/.Line[pts_,rest___]:>Tube[pts,0.005,rest],ParametricPlot3D[
ζ
1
[t],{t,0,1},PlotStyleBlack]/.Line[pts_,rest___]:>Tube[pts,0.005,rest],Graphics3D[{Black,Thick,Line[{{0,0,0},{0,0,2}}]}],Graphics3D[{Black,Thick,Line[{{0,0,2},{2,0,0}}]}]},AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRange{{0,2},{0,1},{0,2}},ViewPoint{1.3,2.4,1.5},ImageSize700],Style["Volume between an elliptic cylinder and plane:",Bold,Large],{{t1,0.00002,"Value (t1)"},0.00001,2,0.00001},ControlPlacementTop
Out[]=
​
Volume between an elliptic cylinder and plane:
Value (t1)