13.3 Notes (8th).pdf - 1 13.3 Arc Length and Curvature DEF...

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1 13.3 Arc Length and Curvature DEF: A parametrization 𝐫𝐫 ( 𝑡𝑡 ) is said to be smooth on an interval 𝐼𝐼 iff 𝐫𝐫′ ( 𝑡𝑡 ) is continuous and 𝐫𝐫′ ( 𝑡𝑡 ) ≠ 𝟎𝟎 on 𝐼𝐼 . A space curve is said to be smooth if it has a smooth parametrization. Let 𝐫𝐫 ( 𝑡𝑡 ) = 〈𝑥𝑥 ( 𝑡𝑡 ), 𝑦𝑦 ( 𝑡𝑡 ) be a smooth parametrization of a curve that is traversed only once from 𝑡𝑡 = 𝑎𝑎 to 𝑡𝑡 = 𝑏𝑏 . We want to find the length 𝐿𝐿 of this curve. We can use a typical element argument to derive a formula. 𝑑𝑑𝑑𝑑 2 = 𝑑𝑑𝑥𝑥 2 + 𝑑𝑑𝑦𝑦 2 𝑑𝑑𝑑𝑑 = �𝑑𝑑𝑥𝑥 2 + 𝑑𝑑𝑦𝑦 2 𝑑𝑑𝑑𝑑 = 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 2 + 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 2 𝑑𝑑𝑡𝑡 𝑑𝑑𝑑𝑑 = | 𝐫𝐫′ ( 𝑡𝑡 )| 𝑑𝑑𝑡𝑡 𝐿𝐿 = 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 2 + 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 2 𝑏𝑏 𝑎𝑎 𝑑𝑑𝑡𝑡 = | 𝐫𝐫′ ( 𝑡𝑡 )| 𝑑𝑑𝑡𝑡 𝑏𝑏 𝑎𝑎 EXAMPLE: Find the length of the curve defined by 𝐫𝐫 ( 𝑡𝑡 ) = 2 𝑡𝑡 , 𝑡𝑡 2 , 1 3 𝑡𝑡 3 ) , 0 ≤ 𝑡𝑡 ≤ 1 . 𝐫𝐫′ ( 𝑡𝑡 ) = 2,2 𝑡𝑡 , 𝑡𝑡 2 ) | 𝐫𝐫′ ( 𝑡𝑡 )| = 4 + 4 𝑡𝑡 2 + 𝑡𝑡 4 | 𝐫𝐫′ ( 𝑡𝑡 )| = ( 𝑡𝑡 2 + 2) 2 | 𝐫𝐫′ ( 𝑡𝑡 )| = | 𝑡𝑡 2 + 2| | 𝐫𝐫′ ( 𝑡𝑡 )| = 𝑡𝑡 2 + 2 since 𝑡𝑡 2 + 2 0 𝐿𝐿 = | 𝐫𝐫′ ( 𝑡𝑡 )| 1 0 𝑑𝑑𝑡𝑡 = ( 𝑡𝑡 2 + 2) 1 0 𝑑𝑑𝑡𝑡 = 1 3 𝑡𝑡 3 + 2 𝑡𝑡� | 0 1 𝐿𝐿 = 1 3 + 2 � − 0 = 7 3
2 EXAMPLE: Find the length of the curve defined by 𝐫𝐫 ( 𝑡𝑡 ) = 〈𝑐𝑐𝑐𝑐𝑑𝑑𝑡𝑡 , 𝑑𝑑𝑠𝑠𝑠𝑠𝑡𝑡 , 𝑡𝑡 ) , 0 ≤ 𝑡𝑡 ≤ 2 𝜋𝜋 . | 𝜋𝜋

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