RevolutionSurfaceArea::usage=" Find the surface area of the surface generated by rotating the curve x =
2
y
+ 2y, -1 ≤ y ≤ 1 with respect to the Y axis. References: Personal.";
In[]:=
Manipulate[ClearAll;Show[{RevolutionPlot3D[{
2
y
+2y,y,0},{y,-1,1},{θ,0,θ1},RevolutionAxis{0,1,0},PlotStyle->Opacity[0.6],MeshFalse,BoundaryStyleDirective[Black,Thick],PerformanceGoal"Quality",ColorFunction"BlueGreenYellow"],ParametricPlot3D[t{0,1,0},{t,-3,3},ColorFunction->"BlueGreenYellow"],Graphics3D[{Black,Ball[{0,1,0},0.07]}],Graphics3D[{Black,Ball[{3Cos[-θ1],1,3Sin[-θ1]},0.07]}],​​Graphics3D[{Black,Ball[{0,-1,0},0.07]}],Graphics3D[{Black,Ball[{-Cos[-θ1],-1,-Sin[-θ1]},0.07]}],​​Graphics3D[{Black,Dashed,Thick,Line[{{0,1,0},{3Cos[-θ1],1,3Sin[-θ1]}}]}],Graphics3D[{Black,Dashed,Thick,Line[{{0,-1,0},{-Cos[-θ1],-1,-Sin[-θ1]}}]}]},AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,ViewPoint{1.3,1,2.},ImageSizeFull],Style["Surface area:",Bold,Large],{{θ1,1,"Value (θ)"},0.1,2π,0.1},ControlPlacementTop]
Out[]=
​
Surface area:
Value (θ)
Surface area of revolution:
In[]:=
f[y_]:=
2
y
+2y;
In[]:=
A=N
1
∫
-1
2πf[y]
1+
2
(f'[y])
y
Out[]=
24.0294
In[]:=
ContourPlot3D-
2
x
+
2
z
==
2
y
+2y,
2
x
+
2
z
==
2
y
+2y,{x,-3,3},{y,-1,1},{z,-3,3},BoxRatios->Automatic
Out[]=
It is found that the equations of the surfaces of revolution are:
-
2
x
+
2
z
=
2
y
+2y,
2
x
+
2
z
=
2
y
+2y
.