Math 2414 Section 6 5 Length of Curves Arc Length y f x Suppose a curve is given by where f is a function with a continuous first derivative a b on the interval The goal is to determine how far you would travel if you walked along the a f a b f b curve from to This distance is the arc length which we denote L Use the distance formula and the Mean Value Theorem to determine a formula for arc length Definition Arc Length for y f x a b Let f have a continuous first derivative on the interval The length of the curve from a f a to b f b is b 2 L 1 f x dx a I will expect you to list ALL points of continuity You cannot assume that you will always be given functions with continuous derivatives Find the arc length of the curve f x x 3 2 between x 0 and x 4 1 f x 2e x e x 0 ln 2 8 Find the length of the curve on the interval Definition Arc Length for x g y x g y c d Let have a continuous first derivative on the interval The length of the curve g c c g d d from to is d 2 L 1 g y dy c Find the length of the curve x 2 y 4 on the interval 3 y 4 Find the arc length of the curve x e y e y 0 y ln 2 on the interval 2 1
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