VolumeSurfaceUnitaryDisk::usage="Find the volume of the solid below the surface: z = 1 - 4
2​
x
2
y
and above of unitary disk. References: Hollis, Selwyn. CalcLabs with Mathematica, Multivariable ​​ Calculus. Cengage 4a. Ed.: p.119, 2010; Personal. ";
In[]:=
ManipulateV=4
1
∫
0
1-
2
x
∫
0
(1-4
2
x
2
y
)yx;​​p=Grid[{{Volume},{V}},FrameAll,ItemSize->15,ItemStyle->15];​​Plot3D[1-4
2
x
2
y
,{x,-1.1,1.1},{y,-1.1,t},BoxRatiosAutomatic,RegionFunctionFunction[{x,y},
2
x
+
2
y
<=1],FillingBottom,PerformanceGoal->"Quality"],{t,-1.09,1.1,0.01},Delimiter,{{p,2,"Volume"}},Delimiter,SaveDefinitions->True,ControlPlacement->Top
Out[]=
​
t
Volume
Volume
5π
6
Another method for volume (suggested by the Ph.D Frank Kampas):
In[]:=
s1=ImplicitRegion[z<=1-4
2
x
2
y
,{x,y,z}];​​s2=ImplicitRegion[
2
x
+
2
y
<=1,{x,y,z}];​​s3=ImplicitRegion[z>=0,{x,y,z}];​​Volume@RegionIntersection[s1,s2,s3]
Out[]=
5π
6