Application of  Integration to Calculate Arc Length

 

In a Nut Shell:  Calculation of arc length, s, is based on the Pythagorean Theorem.
The calculation typically involves three steps as follows:

 

 

Step 1  Visualize a “small” (differential) element, ds,  tangent to the curve, C, at an arbitrary

location  (x,y)  as shown below.  The length  ds   can be calculated using the Pythagorean

theorem.

                                       ds2  =  dx2  +  dy2       Thus    ds  =  √( dx2  +  dy2 )

 

                         

 

                                    

 

 

Step 2    For  y  =  y(x)  Write   ds  using x  as the independent variable.

 

         ds   =    [√ 1  +  (dy/dx)2 ] dx

 

If it happens that  x  is given in terms of y, then   x  =  x(y)  and  ds can be written as:

 

        ds  =  [√ 1  +  (dx/dy)2 ] dy

 

 

Step  3   Determine the limits of integration in order to find the total arc length.  i.e.

 

                a  ≤   x    ≤   b       (or  c  ≤   y    ≤   d  )

 

Perform the integration to find the total arc length. 

 

                           b

              S   =     ∫  [√ 1  +  (dy/dx)2 ] dx                                 Click here for an example.

                           a

 

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