AreaOfRevolution::usage="Find the surface area that is generated by rotating the curve y = Sin[π x], 0 ≤ x ≤ 1 around the X axis. Reference: Personal.";
In[]:=
Manipulate[ClearAll;Show[{RevolutionPlot3D[{x,Sin[πx],0},{x,0,1},{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,-1,1},ColorFunction->"BlueGreenYellow"],Graphics3D[{Black,Ball[{0.5,0,0},0.02]}],Graphics3D[{Black,Ball[{0.5,Cos[θ1],Sin[θ1]},0.02]}],​​Graphics3D[{Black,Dashed,Thick,Line[{{0.5,0,0},{0.5,Cos[θ1],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 (θ)"},0.1,2π,0.1},ControlPlacementTop]
Out[]=
​
Surface of revolution:
Value (θ)
Area of the revolution surface:
In[]:=
f[x_]:=Sin[πx];
In[]:=
A=N
1
∫
0
2πf[x]
1+
2
(f'[x])
x
Out[]=
7.77939
In[]:=
ContourPlot3D
2
y
+
2
z
==Sin[πx],{x,0,1},{y,-1,1},{z,-1,1},BoxRatios->Automatic
Out[]=
The equation of the surface of revolution is:
2
y
+
2
z
=Sin[πx]
.
​