# PP 9.1 - Arc Length We know how to find the distance...

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Arc Length

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We know how to find the distance between two points. That is equivalent to finding the length of the green line segment in the graph. We want to find the distance between the two points on the blue curve. We want to find the arc length of the blue curve between the two red points,
Getting the Formula ion. considerat under arc the of n descriptio the be , ), ( Let b x a x f y = ]. , [ of partition a be ... Let 2 1 0 b a b x x x x a n = < < < < = n i x f x i i = 0 )), ( , ( P Let i

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Increasing the number of points of the partition allows us to approximate the arc length using the distance formula. n = 3 n = 5
Then, ) , ( lim length Arc 1 1 = - = = n i i i n P P d L ( 29 = - - - + - = n i i i i i n x f x f x x 1 2 1 2 1 ) ( ) ( ) ( lim = + = n i i i n y x 1 2 2 ) ( ) ( lim ( 29 [ ] = + = n i i i i n x x f x 1 2 * 2 ) ( lim By the Mean Value Theorem applied for each subinterval.

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( 29 [ ] = + = n i i i i n x x f x L 1 2 * 2 ) ( lim [ ] i n i i n x x f + = = 1 2 * ) ( 1 lim [ ] dx x f b a + = 2 ) ( 1 [ ] dy y g L d y c y g x d c
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