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 timeand 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
so the relevant definite integral is
Lindsay has traveled 225 feet in 15 seconds, for an average speed of 15 feet per
second!