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Unformatted text preview: 14.3 Arc Length and Curvature Recall: If C : x = f ( t ), y = g ( t ) for a ≤ t ≤ b be a parameterized curve, that is traversed once, with f and g continuous on [ a,b ] then the arc length of C is L = b a [ f ( t )] 2 + [ g ( t )] 2 dt. Definition 14.3.1 Now let C be a curve parameterized by x = f ( t ) , y = g ( t ) , z = h ( t ) , a ≤ t ≤ b, where r ( t ) = < f ( t ) ,g ( t ) ,h ( t ) > and r ( t ) is continuous on [ a,b ] . If C is traversed once, then the length of C is given by L = b a [ f ( t )] 2 + [ g ( t )] 2 + [ h ( t )] 2 dt = Example 14.3.1 Find the length of the curve C associated with the vector-valued function r ( t ) = < sin(2 t ) , cos(2 t ) , 2 t > from the point (0 , 1 , 0) to (0 , 1 , 4 π ) . Example 14.3.2 Find the length of the curve C represented by the vector-valued function r ( t ) = < sin t, cos t,t > on the interval ≤ t ≤ 4 π . Question: How does the parametrization affect the arc length of the curve?...
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