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CHAPTER 2

Arc Length. CHAPTER 2. 2.4 Continuity. Arc Length Formula: If a smooth curve with parametric equations x = f ( t ) , y = g ( t ), a  t  b , is traversed exactly once as t increases from a to b, then its length is

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CHAPTER 2

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  1. Arc Length CHAPTER 2 2.4 Continuity Arc Length Formula:If a smooth curve with parametric equations x = f (t),y = g(t), a  t b, is traversed exactly once as t increases from a to b, then its length is L = ab[(dx / dt)2+ (dy / dt)2] 1/2dt. L = ab[ 1 + (dy /dt)2] 1/2dx L = ab[(dx / dt)2+ 1] 1/2dy

  2. Example: Graph the curve and find its exact length. x = 3t – t3, y = 3t2, 0  t  2. Example: Find the total length of the asteroid x = a sin3, y = a cos3. Example: Use table of integrals to find the exact length of the curve x = ln (1- y2) 0 <= y <= 1/2.

  3. Average Value of a Function CHAPTER 2 2.4 Continuity fave = (1/ (b – a))abf (x) dx Mean Value Theorem for Integrals:If f is continuous on [a,b], then there exists a number c in [a,b] such that abf (x) dx = f (c) (b - a).

  4. CHAPTER 2 Example: Find the average value of f on the given interval f (x) = sin x, [0,]. 2.4 Continuity Example: For f (x) = 4x – x2 , [0,3]: a) Find the average value of f on the given interval. b) Find c such that fave= f (c). c) Sketch the graph of f and a rectangle whose area is the same as the area under the graph of f.

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