Now that we know how to compute the velocity (and hence speed) of an object whose
position at time *t* is given by (*x*(*t*), *y*(*t*)), it is only a small step to compute the
distance the object travels over a certain period of time--and hence, the length of a
parametric curve. Let us return to the example given earlier regarding Lindsay's ice
skating along the parametric curve (*x*(*t*), *y*(*t*)) (where *x* and *y* are measured in feet)
from *t* = 0 to *t* = 15 seconds. Suppose Lindsay decides to skate around the rink faster
and faster along a circular path, so her position is given by given

x(t) | = | cos(t^{2}) | |

y(t) | = | sin(t^{2}) |

In order to find the total distance Lindsay travels, we need only integrate her speed from
the time she starts to the time she stops. Her speed at time *t* is given by

= | |||

= | |||

= | |||

= | 2t |

so the relevant definite integral is

dt = 2tdt = 225. |

Lindsay has traveled 225 feet in 15 seconds, for an average speed of 15 feet per second!

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