AllAboutACycloid::usage="5. The cycloid is the curve given parametrically by the equationsx (t) = t − sin t, and y (t) = 1 − cos t for 0 ≤ t ≤ 2π.(a) Sketch the cycloid.(b) Find the arc-length of the cycloid.(c) Find the area bounded by the cycloid and the x-axis.(d) Find the area of the surface of revolution generated by rotating the cycloid around the x-axis.(e) Find the volume enclosed by the surface of revolution generated by rotating the cycloid around the x-axis.References: Internet: maths.ox.ac.uk, Geometry Exercises, Ejercicio 5; Personal ";
(a) Graph of the cycloid:
In[]:=
Manipulate[​​gr=ParametricPlot[{t-Sin[t],1-Cos[t]},{t,0,t1},PlotStyleThick,ColorFunction->"Aquamarine"];​​g2=Circle[{t,1},1]/.tt1-0.01;​​g3=Point[{t-Sin[t],1-Cos[t]}]/.tt1-0.01;​​g4=Line[{{t,1},{t-Sin[t],1-Cos[t]}}]/.tt1-0.01;​​g5=Point[{t,1}]/.tt1-0.01;​​Show[{gr,Graphics[{Red,Thick,PointSize[0.018],g3}],Graphics[{Black,Thick,PointSize[0.018],g5}],Graphics[{Purple,Thick,g2}],Graphics[{Blue,Thick,Dashed,g4}]},AxesTrue,AxesOrigin{0,0},BoxRatiosAutomatic,PlotRangeAll,PerformanceGoal"Quality",ImageSizeFull],Style["Cycloid:",Bold,Medium],{{t1,1,"Angle (θ)"},0.01,2π,0.1},ControlPlacementTop]
Out[]=
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Cycloid:
Angle (θ)
(b) Arc length:
In[]:=
x[t_]:=t-Sin[t];y[t_]:=1-Cos[t];
In[]:=
ℒ=
2π
∫
0
2
(x'[t])
+
2
(y'[t])
t
Out[]=
8
(c) Area under the curve:
In[]:=
A=
2π
∫
0
y[t]x'[t]t
Out[]=
3π
(d) Area of the revolution surface around the X axis:
In[]:=
Manipulate[ClearAll;Show[{RevolutionPlot3D[{t-Sin[t],1-Cos[t],0},{t,0,2π},{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,-1,8},ColorFunction->"BlueGreenYellow"],Graphics3D[{Black,Ball[{π,0,0},0.08]}],Graphics3D[{Black,Ball[{π,2Cos[θ1],2Sin[θ1]},0.08]}],​​Graphics3D[{Black,Dashed,Thick,Line[{{π,0,0},{π,2Cos[θ1],2Sin[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["Revolution surface:",Bold,Medium],{{θ1,1,"Value (θ)"},0.1,2π,0.1},ControlPlacementTop]
Out[]=
​
Revolution surface:
Value (θ)
In[]:=
A=
2π
∫
0
2πy[t]
2
(x'[t])
+
2
(y'[t])
t
Out[]=
64π
3
(e) Volume of the revolution surface:
In[]:=
V=
2π
∫
0
π
2
(y[t])
x'[t]t
Out[]=
5
2
π