VolumeExponentialCylinder::usage="Calculate the volume using double integrals, of the surface S: z = 4
4
2
x

,z ≥ 0 D: 2y ≤ x ≤ 2, 0 ≤ y ≤ 1. References: Ronny online, YouTube; Personal. ";​​​​ContourPlot3Dz==4
4
2
x

,x==2,x==2y,z==0,y==0,{x,0,2},{y,0,1},{z,0,10},AxesTrue,ContourStyleDirective[Opacity[0.5],Specularity[White,30]],MeshFalse,PlotRangeAll,AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),BoxRatiosAutomatic,AxesOrigin{0,0,0},BoxedFalse​​​​
Out[]=
ManipulateV=Simplify
1
∫
0
2
∫
2y
4
4
2
x

xy;​​p=Grid[{{Volume},{V}},FrameAll,ItemSize->15,ItemStyle->15];​​Showcurv1=ParametricPlot3Du,
u
2
,0,{u,0,u1},BoundaryStyleDirective[Thick,Black],​​curv2=ParametricPlot3D[{2,u,0},{u,0,u2},BoundaryStyleDirective[Thick,Black]],​​curv3=ParametricPlot3D[{u,0,0},{u,0,u1},BoundaryStyleDirective[Thick,Black]],​​curv4=ParametricPlot3Du,0,4
2
u

,{u,0,u1},BoundaryStyleDirective[Thick,Black],​​curv5=ParametricPlot3D[{2,u,4
4

},{u,0,u2},BoundaryStyleDirective[Thick,Black]],​​curv6=ParametricPlot3D[{2,0,u},{u,0,u3},BoundaryStyleDirective[Thick,Black]],​​curv7=ParametricPlot3D[{2,0,u},{u,0,u3},BoundaryStyleDirective[Thick,Black]],​​curv8=ParametricPlot3D[{2,1,u},{u,0,u3},BoundaryStyleDirective[Thick,Black]],​​curv9=ParametricPlot3Du,
u
2
,4
2
u

,{u,0,u1},BoundaryStyleDirective[Thick,Black],​​surf1=ParametricPlot3Du,s
u
2
,0,{u,0,u1},{s,0,t1},MeshFalse,PlotStyleDirective[Yellow,Opacity[0.5],Specularity[White,20]],​​surf2=ParametricPlot3Du,
u
2
,4s
2
u

,{u,0,u1},{s,0,t1},MeshFalse,PlotStyleDirective[Brown,Opacity[0.5],Specularity[White,20]],​​surf3=ParametricPlot3Du,s
u
2
,4
2
u

,{u,0,u1},{s,0,t1},MeshFalse,PlotStyleDirective[Red,Opacity[0.5],Specularity[White,20]],​​surf4=ParametricPlot3Du,0,4s
2
u

,{u,0,u1},{s,0,t1},MeshFalse,PlotStyleDirective[Blue,Opacity[0.5],Specularity[White,20]],​​surf5=ParametricPlot3D[{2,u,4s
4

},{u,0,u2},{s,0,t1},MeshFalse,PlotStyleDirective[Pink,Opacity[0.5],Specularity[White,20]]],​​AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,MeshFalse,PlotRange10,PerformanceGoal"Quality",ViewPoint{1.3,-2.4,2.},​​{u1,0.1,2,0.1},Delimiter,{u2,0.1,1,0.1},Delimiter,{u3,0.1,4
4

,0.1},Delimiter,{t1,0.1,1,0.1},Delimiter,{{p,2,"Volume"}},SaveDefinitions->True,ControlPlacement->Top
Out[]=
​
u1
u2
u3
t1
Volume
Volume
1
4
-1+
16


Another method to obtain the volume (suggested by the Ph.D Frank Kampas):
In[]:=
s1=ImplicitRegion0<=z<=4
4
2
x

,{x,y,z};​​s2=ImplicitRegion[2y≤x≤2,{x,y,z}];​​s3=ImplicitRegion[0≤y≤1,{x,y,z}];​​Volume@RegionIntersection[s1,s2,s3]
Out[]=
1
4
(-1+
16

)