CHAPTER Continuity Arc Length 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 =  a b [(dx / dt) 2 + (dy / dt) 2 ] 1/2 dt. L =  a b [ 1 + (dy /dt) 2 ] 1/2 dx L =  a b [(dx / dt) 2 + 1] 1/2 dy

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

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

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