Arc Length

Recall finding arc length in Cartesian coordinates. The idea was to divide the curve into small segments, calculate the length of each one, add them up, and take a limit. We determined that

where the length element has the structure

To use as a variable, we factored out the :

However, if we want to use as a variable of integration, we just have to introduce a differential in the same way:

since we have :

so we have described the curve using two functions of the single variable (parameterization).
Now we differentiate and simplify:

adding these two together gives