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
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
|dt = 2tdt = 225.|
Lindsay has traveled 225 feet in 15 seconds, for an average speed of 15 feet per second!
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