VolumeHyperbolicParaboloidCircled::usage="Determine the volume of the region in the first octant under the surface z = x y on the plane XY that is within the unitary circle with center at the origin and to the right of the line x + y = 1. References: Internet: Ronny on line; Personal.";
Development for volume:
Method 1:
In[]:=
V=
1
∫
0
1-
2
x
∫
1-x
xyyx
Out[]=
1
12
Method 2 (suggested by the Ph.D Frank Kampas):
In[]:=
s1=ImplicitRegion[z<=xy,{x,y,z}];​​s2=ImplicitRegion1-x<=y<=
1-
2
x
,{x,y,z};​​s3=ImplicitRegion[z>=0,{x,y,z}];​​Volume@RegionIntersection[s1,s2,s3]
Out[]=
1
12
Development for graph:
Using my notebook Maxymin:
In[]:=
Maxymin[xy,{-5,5},{-5,5}]
Saddle point: {0,0,0}
Out[]=
Null
When x = y in
2
x
+
2
y
=1,thenx=
1
2
andz=xy,theresultisz=
1
2
.
Animation:
In[]:=
Manipulatef[x_,y_]:=xy;ShowPlot3Df[x,y],{x,0,1},{y,0,2t1},BoxRatiosAutomatic,RegionFunctionFunction{x,y},1-x<=y<=
1-
2
x
,FillingBottom,PerformanceGoal->"Quality",PlotStyleDirective[Orange,Opacity[0.6],Specularity[White,40]],AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,PlotRange{{0,1},{0,1},{0,0.5}},ViewPoint{1.3,-2.4,0.5},ImageSize700,Style["Volume of a vaulted solid between hyperbolic paraboloid and delimited circle:",Bold,Large],{t1,0.00002,"Value (t1)"},0.00001,
1
2
,0.00001,ControlPlacementTop
Out[]=
​
Volume of a vaulted solid between hyperbolic paraboloid and delimited circle:
Value (t1)