VolumeParaboloidParabola::usage="Determine the volume of the region under the surface z =
2
x
on the plane XY that is within the parabola y = 2 -
2
x
and to the the line y = x. References: Internet: Ronny on line; Personal.";
Development for volume:
Method 1:
In[]:=
V=
1
∫
-2
2-
2
x
∫
x
2
x
yx
Out[]=
63
20
Method 2 (suggested by the Ph.D Frank Kampas):
In[]:=
s1=ImplicitRegion[z<=
2
x
,{x,y,z}];​​s2=ImplicitRegion[x<=y<=2-
2
x
,{x,y,z}];​​s3=ImplicitRegion[z>=0,{x,y,z}];​​Volume@RegionIntersection[s1,s2,s3]
Out[]=
63
20
Development for graph:
Using my notebook Maxymin:
In[]:=
Maxymin[
2
x
,{-5,5},{-5,5}]
Out[]=
Null
In[]:=
Solve[{y==x,y==2-
2
x
},{x,y}]
{{x-2,y-2},{x1,y1}};
Animation:
In[]:=
Manipulate[f[x_,y_]:=
2
x
;Show[{Plot3D[f[x,y],{x,-2,1},{y,-2,t1},BoxRatiosAutomatic,RegionFunctionFunction[{x,y},x<=y<=2-
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{{-2,1},{-2,1},{0,4}},ViewPoint{1.3,2.4,2},ImageSize700],Style["Volume of a vaulted solid between parabolic cylinder and delimited parabola:",Bold,Large],{{t1,-2+0.00002,"Value (t1)"},-2+0.00001,1,0.00001},ControlPlacementTop]
Out[]=
​
Volume of a vaulted solid between parabolic cylinder and delimited parabola:
Value (t1)