VolumenBóvedaDeViviani::usage="Use double integrals and polar coordinates to find the volume of the solid delimited by: z =
16-
2
x
-
2
y
and
2
x
+
2
y
- 4 x = 0, delimited below by the plane z = 0. References: Ronny online, YouTube; Personal. ";​​​​ContourPlot3Dz==
16-
2
x
-
2
y
,
2
x
+
2
y
-4x==0,z==0,{x,-4.1,4.1},{y,-4.1,4.1},{z,0,6},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[]=
In[]:=
ManipulateV=Simplify2
π/2
∫
0
4Cos[θ]
∫
0
16-
2
r
rrθ;​​p=Grid[{{Volume},{V}},FrameAll,ItemStyle->15];​​Showsurf1=ParametricPlot3D{Abs[4Cos[π(u+v)]Cos[π(u-v)]],4Sin[π(u+v)]Cos[π(u-v)],Abs[4Sin[π(u-v)]]},u,
1
2
,
3
4
,v,0,
1
2
,MeshFalse,PlotStyleDirective[Green,Opacity[0.5],Specularity[White,20]],MaxRecursion->3,​​surf2=ParametricPlot3D2+2Cos[u],2Sin[u],s
8-8Cos[u]
,{u,0,2π},{s,0,t1},MeshFalse,PlotStyleDirective[LightRed,Opacity[0.5],Specularity[White,20]],MaxRecursion->3,​​surf3=ParametricPlot3D[{2+2Cos[u],2sSin[u],0},{u,0,2π},{s,0,t1},MeshFalse,PlotStyleDirective[Yellow,Opacity[0.5],Specularity[White,20]],MaxRecursion->3],​​curv1=ParametricPlot3D[{2+2Cos[u],2Sin[u],0},{u,0,2π},BoundaryStyleDirective[Thick,Black]],curv2=ParametricPlot3D2+2Cos[u],2Sin[u],
8-8Cos[u]
,{u,0,2π},BoundaryStyleDirective[Thick,Black],AxesTrue,AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),BoxRatiosAutomatic,AxesOrigin{0,0,0},PerformanceGoal"Quality",BoxedFalse,PlotRangeAll,​​{t1,0.1,1,0.1},Delimiter,{θ1,0.1,2π,0.1},Delimiter,{{p,2,"Volume of Viviani's vault"}},SaveDefinitions->True,ControlPlacement->Top
Out[]=
​
t1
θ1
Volume of Viviani's vault
Volume
64
9
(-4+3π)
Another way to get the volume suggested by the Ph.D Frank Kampas is:
In[]:=
s1=ImplicitRegion[
2
x
+
2
y
+
2
z
<=16,{x,y,z}];​​s2=ImplicitRegion[
2
x
+
2
y
-4x<=0,{x,y,z}];​​s3=ImplicitRegion[z<=0,{x,y,z}];​​Volume@RegionIntersection[s1,s2,s3]
Out[]=
64
9
(-4+3π)