LengthOfACurve::usage="Find the length of the curve t,​
3
t
3
2
a
​,​
2
a
2t
​ between the planes y = a and y = 3 a.References: Fedenko, A.S. Problemas de Geometría Diferencial. A.S. Editorial Mir, Moscú: p.65, 1981; Personal.";
Development:
In[]:=
ζ[t_]:=t,
3
t
3
2
a
,
2
a
2t
;​​x[t_]:=t;​​y[t_]:=
3
t
3
2
a
;​​z[t_]:=
2
a
2t
;​​
3a
∫
a
2
x'[t]
+
2
y'[t]
+
2
z'[t]
t​​
Out[]=
9a
Let a = 1:
In[]:=
​​ManipulateClearAll;ζ[t_]:=t,
3
t
3
,
1
2t
;​​ShowContourPlot3Dy==
1
3
,{x,-5,5},{y,-5,5},{z,-5,5},ContourStyle->{Opacity[0.1],Green},MeshFalse,PerformanceGoal"Quality",​​ContourPlot3D[y==9,{x,-5,5},{y,-10,10},{z,-5,5},ContourStyle->{Opacity[0.1],Yellow},MeshFalse,PerformanceGoal"Quality"],​​ParametricPlot3D[ζ[t],{t,1,t1},PlotStyle{Thick,Blue}]​​,AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRange10,ViewPoint{-3,3,15},ImageSize500,Style["Arc length:",Bold,Large],{{t1,1.1,"Value (t1)"},1+0.01,3,0.01},ControlPlacementTop
Out[]=
​
Arc length:
Value (t1)