Parametric and Polar Curves
Length of a Parametric Curve
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
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= |
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| = |
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| = |
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| = | 2t |
so the relevant definite integral is
dt = 2tdt = 225.
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Lindsay has traveled 225 feet in 15 seconds, for an average speed of 15 feet per second!
