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Calculus II (Notes) / Parametric Equations and Polar Coordinates / Arc Length with Polar Coordinates   [Notes] [Practice Problems] [Assignment Problems]

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Paul
August 7, 2018


Calculus II - Notes
Applications of Integrals Previous Chapter   Next Chapter Series & Sequences
Area with Polar Coordinates Previous Section   Next Section Surface Area with Polar Coordinates

 Arc Length with Polar Coordinates

We now need to move into the Calculus II applications of integrals and how we do them in terms of polar coordinates.  In this section we’ll look at the arc length of the curve given by,

 

 

where we also assume that the curve is traced out exactly once.  Just as we did with the tangent lines in polar coordinates we’ll first write the curve in terms of a set of parametric equations,

 

 

 

and we can now use the parametric formula for finding the arc length.

 

We’ll need the following derivatives for these computations.

 

 

We’ll need the following for our ds.

 

 

 

The arc length formula for polar coordinates is then,

 

                                                                   

where,

                                                        

 

Let’s work a quick example of this.

 

Example 1  Determine the length of  .

 

Solution

Okay, let’s just jump straight into the formula since this is a fairly simple function.

                                                            

 

We’ll need to use a trig substitution here.

                                            

                                      

                                    

 

The arc length is then,

                                            

 

Just as an aside before we leave this chapter.  The polar equation  is the equation of a spiral.  Here is a quick sketch of  for .

 

PolarArcLength_Ex1_G1

Area with Polar Coordinates Previous Section   Next Section Surface Area with Polar Coordinates
Applications of Integrals Previous Chapter   Next Chapter Series & Sequences

Calculus II (Notes) / Parametric Equations and Polar Coordinates / Arc Length with Polar Coordinates    [Notes] [Practice Problems] [Assignment Problems]

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