8.1 Arc Length - 8.1 Arc Length Definition If the function...

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8.1 Arc Length Definition: If the function given by represents a smooth curve on the interval [ a , b ], then the arc length of f between a and b is given by Similarly, for a smooth curve given by , the arc length of g between c and d is given by ) ( x f y = [ ] dx dx dy dx x f L b a b a + = + = 2 2 1 ) ( ' 1 [ ] dy dy dx dy y g L d c d c + = + = 2 2 1 ) ( ' 1 ) ( y g x =
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Examples: 1) Find the arc length of the graph of from point A (0,1) to point B (8,5). Solutions: ( 29 2 3 1 x y = - ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 1 10 10 27 8 4 40 27 1 5 9 3 2 9 1 2 1 5 9 2 1 4 5 9 1 4 5 9 1 4 9 1 1 1 2 3 1 2 / 3 3/2 5 1 2 / 3 2 / 1 5 1 5 1 2 2 2 / 1 2 / 3 - = - = - = - = - = + = - = - + = + - = - ± = y dy y dy y dy dy dx L y y dy dx y dy dx y x d c Choosing the positive value of x , we have
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2) Use the arc length formula to find the length of the curve Check your answer by noting that the curve is a quarter-circle. Solutions: 2 0 , 4 2 - = x x y ( 29 ( 29 ( 29 [ ] π θ θ θ θ θ θ θ θ θ π θ θ θ π θ π θ π π π π = = = = - = = = = - = - = - = + = - = - + - - = - + = + - = - - = - = - 2 / 0 2 / 0 2 / 0 2 / 0 2 2 0 2 2 0 2 2 2 2 2 2 2 2 2 2 2 2 / 1 2 2 / 1 2 2 2 cos cos 2 4 sin 4 4 cos 2 2 0 : Limit Lower 2 / : Limit Upper cos 2 2 / 2 / , sin 2 let 4 1 2 4 4 1 4 4
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  • Spring '08
  • varies
  • Calculus, Arc Length, Limit of a function, Limit of a sequence, Limit superior and limit inferior, dx

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