Calculus/Parametric Integration

Parametric Differentiation Parametric and Polar Equations/Parametric Equations/Exercises Introduction 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, Example 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 Volume