Lecture_5 - CALCULUS II Spring 2009 LECTURE 5 This lecture...

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CALCULUS II Spring 2009 LECTURE 5 This lecture provides a definition of arc length, an integral model of arc length and several examples of arc length calculations. 1
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Recall We recall from Lecture 4 the notion of an improper integral. End of Recall 2
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Arc Length Our goal is to (i) use human intuition to make a precise mathematical defini- tion of the term arc length of a planar curve 1 (ii) note that the definition models the human notion of arc length for continuous curves and then (iii) find a computable mathematical expression that yields arc length (a number) for planar curves whose derivatives are contin- uously differentiable (or piecewise continuously differentiable). We assume that a curve C is modeled by a real-valued function y ( x ) defined on an interval [ a, b ]. (We imagine a continuous curve, but continuity is not necessary for our definition.) One partitions the curve by choosing points ( x i , y i ) , 0 i n on the curve, so that x i - 1 < x i , 1 i n . One (intuitively) approximates the curve C by a piecewise linear curve, con- sisting of n chords; each chord is determined by two planar points ( x i - 1 , y i - 1 ) and ( x i , y i ) , 1 i n . (We write y i for y ( x i ).) Since we have already de- fined the notion of the length of a chord in the plane one can model the length of the piecewise linear curve by summing the lengths of the chords, that is, by n X i =1 p ( x i - x i - 1 ) 2 + ( y i - y i - 1 ) 2 . One then uses this sum of chord lengths as an approximation for what one intuitively believes the length of C should be. It is noted that this ap- proximation depends upon both the number and placement of the partition points { ( x i , y i ) } n i =0 . One also notes that if one more point is adjoined to a given partition then the sum of the lengths of the chords will not decrease 1 One recalls that a planar curve is a point set in R 2 that is a real-valued function defined on some interval. 3
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and almost always, will increase. This suggests considering partitions with any (finite) number of points and defining the length of the curve as the least upper bound of all possible sums of the chord lengths. In what follows we partition the interval [ a, b ] instead of the curve itself. This is done without loss since it is assumed the the curve is modeled by a real-valued function defined on [ a, b ]. Definition . The arc length of a curve C , modeled by a real-valued function, y ( x ), defined on the interval [ a, b ] is defined to be the least upper bound of n X i =1 p ( x i - x i - 1 ) 2 + ( y i - y i - 1 ) 2 where the least upper bound is computed over all finite partitions of [ a, b ] and where y i = y ( x i ) , 0 i n . It is noted that this least upper bound could be . That is, there exist continuous real-valued functions defined on [ a, b ] whose graphs have infinite length. One observes that if the curve C is translated or reflected through one of the coordinate axes then the arc length doesn’t change.
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