Arclength
To find the arclength of a curve 
, chop it up into infinitely small pieces of length 
as follows:
To find the arclength of a curve 
, chop it up into infinitely small pieces of length as follows:
Then add up all the lengths of the pieces to get the length of the whole curve: 
.
When you evaluate this integral, you get the arclength 
.
It is clear from the diagrams that the polar coordinate version is way harder than the cartesian coordinate version, so just work in terms of 
and convert to 
at the end.
To get the length of the section of curve between 
and 
, just add up the pieces between and : ![\[\int_{x=a}^b ds=\int_a^b ... dx.\]](http://latex.artofproblemsolving.com/2/e/9/2e9d429515883d0887b9635297e18e80c0bb68da.png)
To get the length of the section of curve between 
and 
, just add up the pieces between and : ![\[\int_{\theta=a}^b ds=\int_a^b ... d\theta.\]](http://latex.artofproblemsolving.com/1/1/1/1118dd7046abdc528305bbbcda7aae4de4a26dd9.png)
