VolumeBetweenParaboloids::usage="Calculate the volume of the solid that is limited above by: z = 1 - - 4 and below by: + 4 - 4 z = 1. References: Internet: Ronny on line; Personal.";
2
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2
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Devolvement for volume:
Method 1:
In[]:=
V=zxy
1
2
∫
-
1
2
1-4
2
y
∫
-
1-4
2
y
1--4
2
x
2
y
∫
+4-14
2
x
2
y
Out[]=
5π
16
Method 2 (suggested by the Ph.D Frank Kampas):
In[]:=
s1=ImplicitRegion[z<=1--4,{x,y,z}];s2=ImplicitRegion[+4-4z<=1,{x,y,z}];Volume@RegionIntersection[s1,s2]
2
x
2
y
2
x
2
y
Out[]=
5π
16
Development for graph:
In[]:=
Solve[{z==1--4,+4-4z==1.},{x,z}]
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2
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Out[]=
x-1.
1.-4.
,z0.,x2
y
1.-4.
,z0.2
y
In[]:=
Reduce[1-4>=0,y]
2
y
Out[]=
-≤y≤
1
2
1
2
Using my notebook Maxymin:
In[]:=
Maxymin[1--4,{-5,5},{-5,5}]Maxymin+-,{-5,5},{-5,5}
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2
x
4
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1
4
Maximum: {0,0,1}
Out[]=
Null
Minimum: 0,0,-
1
4
Out[]=
Null
Animation:
In[]:=
Manipulateζ[t_]:=-[t_]:=-4,{x,-1,1},y,-,,{z,0,t1},Mesh->None,MaxRecursion->3,PlotStyleDirective[Orange,Opacity[0.6],Specularity[White,40]],RegionPlot3D+4-4z<=1,{x,-1,1},y,-,,z,-,t1-1,Mesh->None,MaxRecursion->3,PlotStyleDirective[Blue,Opacity[0.4],Specularity[White,30]],ParametricPlot3Dζ[t],t,-,,PlotStyle->Black/.Line[pts_,rest___]:>Tube[pts,0.008,rest],ParametricPlot3D[t],t,-,,PlotStyle->Black/.Line[pts_,rest___]:>Tube[pts,0.008,rest],AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRange{-1,1},-,,-,1,ViewPoint{1.3,-2.4,0.5},ImageSize700,Style["Volume between two paraboloids:",Bold,Large],{{t1,0.00002,"Value (t1)"},0.00001,1,0.00001},ControlPlacementTop
1-4
,t,0;2
t
ζ
1
1-4
,t,0;ShowRegionPlot3Dz<=1-2
t
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1
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1
4
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ζ
1
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1
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1
4
Out[]=
