AreaOfRevolutionSurface::usage="Get the area of the surface of revolution when it rotates around the X axis, the function: f(x) = Log[x + 2], 2 ≤ x ≤ 4. References: Internet: matematicaenlinea.com; Personal.";
In[]:=
Manipulate[ClearAll;Show[{RevolutionPlot3D[{x,Log[x+2],0},{x,2,4},{θ,0,θ1},RevolutionAxis{1,0,0},PlotStyle->Opacity[0.6],MeshFalse,BoundaryStyleDirective[Black,Thick],PerformanceGoal"Quality",ColorFunction"BlueGreenYellow"],ParametricPlot3D[t{1,0,0},{t,-1,6},ColorFunction->"BlueGreenYellow"],Graphics3D[{Black,Ball[{2,0,0},0.07]}],Graphics3D[{Black,Ball[{2,Log[4]Cos[θ1],Log[4]Sin[θ1]},0.07]}],​​Graphics3D[{Black,Ball[{4,0,0},0.07]}],Graphics3D[{Black,Ball[{4,Log[6]Cos[θ1],Log[6]Sin[θ1]},0.07]}],Graphics3D[{Black,Thick,Dashed,Line[{{2,0,0},{2,Log[4]Cos[θ1],Log[4]Sin[θ1]}}]}],Graphics3D[{Black,Thick,Dashed,Line[{{4,0,0},{4,Log[6]Cos[θ1],Log[6]Sin[θ1]}}]}]​​},AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,ViewPoint{1.3,-2.4,2.},ImageSizeFull],Style["Surface of revolution:",Bold,Large],{{θ1,1,"Value (θ1)"},0.1,2π,0.1},ControlPlacementTop]
Out[]=
​
Surface of revolution:
Value (θ1)
Area of revolution surface:
In[]:=
f[x_]:=Log[x+2];
In[]:=
A=N
4
∫
2
2πf[x]
1+
2
(f'[x])
x
Out[]=
20.5481
In[]:=
ContourPlot3D
2
y
+
2
z
==Log[x+2],{x,2,4},{y,-Log[6],Log[6]},{z,-Log[6],Log[6]},BoxRatios->Automatic
Out[]=
The equation of the surface of revolution is proven to be:
2
y
+
2
z
=Log[x+2]
.
​