Arc Length and The Unit Tangent Vector

Example 13.3.8

r[t_]={t Sin[t]+Cos[t],t Cos[t]-Sin[t]}
v[t_]=r'[t]
Tangent[t_]=Simplify[((v[t])/(Sqrt[v[t].v[t]]))]
arclength=Integrate[(Sqrt[v[t].v[t]]),{t,(Sqrt[2]),2}]

Example 13.3.10

r[t_]={12 Sin[t],-12Cos[t],5t}
r[0]
v[t_]=r'[t]
arclength=Integrate[(Sqrt[v[t].v[t]]),{t,0,-t}]
Solve[arclength==13Pi,t]
r[-Pi]
Integrate[(Sqrt[v[t].v[t]]),{t,0,Pi}]

Example 13.3.12

r[t_]={t Sin[t]+Cos[t],Sin[t]-t Cos[t]}
v[t_]=r'[t]
arclength=Integrate[(Sqrt[v[t].v[t]]),{t,0,t}]

Integrate[(Sqrt[v[t].v[t]]),{t,Pi/2,Pi}]

Example 13.3.18

r[t_]={Cos[4t],Sin[4t],4t}
v[t_]=r'[t]
arclength=Integrate[(Sqrt[v[t].v[t]]),{t,0,Pi/2}]

r[t_]={Cos[t/2],Sin[t/2],t/2}
v[t_]=r'[t]
arclength=Integrate[(Sqrt[v[t].v[t]]),{t,0,4Pi}]

r[t_]={Cos[t],-Sin[t],-t}
v[t_]=r'[t]
arclength=Integrate[(Sqrt[v[t].v[t]]),{t,-2Pi,0}]