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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(t2)  
y(t)=sin(t2)  

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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