CylinderRevolutionArea::usage="Find the equation of the cylinder of revolution whose axis is the line x = 2y = - z, and whose radius is 4. Find the area of your section by the XY plane. References: Usunáriz, U et al.Problemas de Geometría Analítica y Diferencial.​; Personal.";
Development:
Cylinder’s equation:
In[]:=
Expand
2
Norm1,
1
2
,-1{x,y,z}
==
2
4Norm1,
1
2
,-1

In[]:=
Expand
2
-
x
2
+y
+
2
(-x-z)
+
2
y+
z
2
36
5
2
x
4
-xy+2
2
y
+2xz+yz+
5
2
z
4
36;​​5
2
x
+8
2
y
+5
2
z
-4xy+8xz+4yz-144==0;
Area of your section with the XY plane (z = 0):
5
2
x
+8
2
y
-4xy-144==0;
Using my notebook Conicas6.nb:
In[]:=
Conicas[5,-4,8,0,0,-144,{-10,10},{-10,10}]
Los valores de a, b, c,
LL´
y e son:
6
4
2
5
16
3
5
3
Eje mayor en x''
Después de la rotación la ecuación es:
-144+4
´2
x
+9
´2
y
= 0
El ángulo de rotación es:
θ = 26.5651°
Después de la traslación la ecuación es:
-144+4
´´2
x
+9
´´2
y
= 0
Las coordenadas del nuevo origen son:
´´
O
​(0,0)
Out[]=
It is an ellipse, the area of an ellipse is π a b:
In[]:=
Área=π(6)(4)
Out[]=
24π
We are going to find a line parallel to the axis:
A point on the cylinder is:
1,
1
2
-2
5
,
4
5
-1;
A line parallel to the axis would be:
ℒ:1,
1
2
-2
5
,
4
5
-1+t1,
1
2
,-1;
In[]:=
ManipulateShowRevolutionPlot3D1+t,
1
2
+
1
2
t-2
5
,
4
5
-1-t,{t,-5,5},{θ,0,θ1},RevolutionAxis{2,1,-2},PlotStyle->Opacity[0.6],MeshFalse,BoundaryStyleDirective[Black,Thick],PerformanceGoal"Quality",ColorFunction"BlueGreenYellow",​​ContourPlot3D[z==0,{x,-12,12},{y,-12,12},{z,-6,6},ContourStyle->{Opacity[0.9],Yellow},MeshFalse,PerformanceGoal"Quality"],ParametricPlot3D[{2y,y,-2y},{y,-5,5},ColorFunction->"BlueGreenYellow"],ParametricPlot3Dx,
4x+
4608-144
2
x
16
,0,x,-4
2
,4
2
,PlotStyle->{Thickness[0.005],Black},ParametricPlot3Dx,
4x-
4608-144
2
x
16
,0,x,-4
2
,4
2
,PlotStyle->{Thickness[0.005],Black},AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRange10,ViewPoint{-8,-3,3},ImageSize700,Style["Cylinder of revolution:",Bold,Large],{{θ1,1,"Value (θ)"},0.1,2π,0.1},ControlPlacementLeft
Out[]=
​
Cylinder of revolution:
Value (θ)