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Calculus/Parametric Integration

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IntroductionEdit

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 lengthEdit

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,

 

ExampleEdit

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 areaEdit

VolumeEdit