Arc Length of Parametric Curves We’ve talked about the following parametric representation for the circle: x y = = a cos t a sin t We noted that x2 + y 2 = a2 and that as t increases the point (x(t), y(t)) moves around the circle in the counterclockwise direction. t= π 2 t=0 Figure 1: The parametrization (a cos t, a sin t) has a counterclockwise trajectory. We’ll now learn how to compute the arc length of the path traced out by this trajectory; the result should match our previous result for the arc length of a circular curve. Recall our basic relationship: � ds2 = dx2 + dy 2 or ds = dx2 + dy 2 . We incorporate parameter t into this formula as follows: �� 2 2 dx dy ds = + dt. dt dt So, to compute the inﬁnitesimal arc length ds we start by computing dy dt : dx = −a sin t and dt dy = a cos t. dt Hence, ds = � (−a sin t)2 + (a cos t)2 dt 1 dx dt and = � √ = ds = a2 (sin2 t + cos2 t) dt a2 · 1 dt a dt From this we conclude that the speed at which the point moves around the circle ds is: = a. Because the speed is constant, we say that the point is moving with dt uniform speed. Parametrizations such as: x y = = a cos kt a sin kt are common in math and physics classes. Again this is a parametrization of the circle, but this time the point is moving with uniform speed ak. (We’ll assume that both a and k are positive.) 2