Because most parametric equations are given in explicit form, they can be integrated like many other equations. Integration has a variety of applications with respect to parametric equations, especially in kinematics and vector calculus.
Say that is increasing on some interval, . Recall, as we have derived in a previous chapter, that the length of the arc created by a function over an interval, , is given by,
It may assist your understanding, here, to write the above using Leibniz's notation,
Using the chain rule,
We may then rewrite ,
Hence, becomes,
Extracting a factor of ,
As is increasing on , , and hence we may write our final expression for as,
Take a circle of radius , which may be defined with the parametric equations,
As an example, we can take the length of the arc created by the curve over the interval . Writing in terms of ,
Computing the derivatives of both equations,
Which means that the arc length is given by,
By the Pythagorean identity,
One can use this result to determine the perimeter of a circle of a given radius. As this is the arc length over one "quadrant", one may multiply by 4 to deduce the perimeter of a circle of radius to be .