Arc Length of Curves

 

Motivation:

                         The derivative of a function at a point is the slope of the tangent line; the integral also has its geometry meaning as area enclosed by the curve. It is natural for us to think about the length of the curve.

 

                         For small variation on x, denoted as  x, there will be a corresponding small variation on y, denoted as  y .  So, the chord length of the secant line intercepted by the curve is   s=  .

 

                                             

 

                        So,  intuitively,   we can have

 

                                

                                      =

 

                        And to sum up those infinitesimal pieces of line segment, we can get the length of the curve

 

                                       =

 

                       This should be the formula for the length of a segment of a curve intuitively.  However, not every curve can be approximated by using the sum of infinitesimal line segments – it has to converge.   To be conservative, it is safe to ask for the existence of the derivative of the function.

 

 

Theorem ( Arc Length ): 

            Let  f(x) be differentiable on [a,b] .  The arc length of  f(x) from x=a to x=b is

 

                                                

 

 

 

Example:   Find the length of the curve E: x² + y² = r²  ,  r> 0

 

       Sol:

                   The curve is composed of two functions:

                                             y=  and   y=-

 

                   It is symmetric.  So, it is only necessary to find the length from one of them.

 

                                              =

 

                          ½ S =   =

 

                                     Let  x=r sin(t)  , t  [ ,  ] .

                                    dx = r cos(t) dt 

                                       

                                          =

 

                               So, 

                                          =  =  r

 

                          Thus,     the length of the curve is  2  r .

 

 

 

Example:  Find the length of the curve f(x)=x²  between x=0 and x=1 .

 

       Sol:

                                f’(x) = 2x

                                S=

                                          Set    x=  tan(t) ,  t  [0,  ]  where  =arctan(2) .

                                           dx =  sec²(t) dt

                                                 = sec(t)

                      So,

                                              S=

 

                           To find the value of this integral needs a little work:

                                           I=

                                            =

                                            = sec(t)tan(t) -

                                            = sec(t)tan(t) -

                                            = sec(t)tan(t) -  +

 

                          Thus, 

                                        2I = sec(t)tan(t) +

                                     I =  sec(t)tan(t) +

 

                                          A=

                                             =

                                             =

                                             = ln|sec(t)+tan(t)|

 

                        I =  sec(t)tan(t) +  ( ln|sec(t)+tan(t)| ) + C

              Hence,

                                 S=

                                   =  sec(  )tan(  ) +  ln| sec(  ) + tan(  ) |

 

                                                tan(  ) = 2

                                                sec(  ) =

 

                          S =  +  ln(  +2)