13.3 Arc Length and Curvature

13.3 Arc Length and Curvature - Math 224 Calculus III 13.3...

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Math 224 – Calculus III 1 13.3 Arc Length and Curvature Recommended Homework: #1-37, 43 Arc Length How do we find the length of a curve in space? In the plane, arc length was derived (calc II) using the distance formula Suppose a curve is defined parametrically by ( ) and ( ) x f t y g t Picture : In space, if we think of a vector function as parametric in 3 : ( ) ( ), ( ), ( ) t f t g t h t r means ( ), ( ) and ( ) x f t y g t z h t Then, the length L of the space curve () t r from ta to tb is given by: L Alternatively we can write: L t dt r . L
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Math 224 – Calculus III 2 Examples: Find the lengths of the following: ( ) 6sin2 ,6cos2 ,5 t t t t r on the interval 0 t 3/2 22 ( ) ( cos ) ( sin ) 3 t t t t t t r i j k on the interval 12 t 2 ( ) , ln ,ln t t t t t r on the interval 25 t
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Math 224 – Calculus III 3 Reparametrization of a Space Curve with Respect to Arc Length The parametric representation of a space curve is not unique: 2 2 4 ( ) cos( ),sin( ),2 t t t t r for 0 t is the same space curve as 2 ( ) cos( ),sin( ),2 s s s s r for 0 s
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13.3 Arc Length and Curvature - Math 224 Calculus III 13.3...

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