Calculus/Parametric Integration

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Parametric Integration

Introduction[edit]

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.

So, taking a simple example, with respect to t:

Arc length[edit]

Consider a function defined by,

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,

Example[edit]

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 .

Surface area[edit]

Volume[edit]