mthsc206-fall-2010-notes-14.3

# mthsc206-fall-2010-notes-14.3 - 14.3 Arc Length and...

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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 0 t 4 π . Question: How does the parametrization a ff ect the arc length of the curve? Definition 14.3.2 Let r ( t ) = < f ( t ) , g ( t ) , h ( t ) > , a t b , be a vector-valued function whose associated curve C is parameterized by x = f ( t ) , y = g ( t ) , z = h ( t ) , a t b, is piecewise smooth and is traversed exactly once on [ a, b ] . The arc length function

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