Calculus/Parametric Integration

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Parametric Integration



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


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,




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