Lecture_9 - CALCULUS II Spring 2009 LECTURE 9 This lecture...

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Unformatted text preview: CALCULUS II Spring 2009 LECTURE 9 This lecture provides: a definition of arc length of C as modeled by r ( t ) an arc length formula: = R b a p ( x ( t )) 2 + ( y ( t )) 2 dt the arc lengths of a circle and of a cycloid a note on how arc length of a curve may be model dependent a definition of a polar curve an arc length formula: = R b a p r 2 ( ) + ( r ( )) 2 d calculation of two spiral arc lengths. 1 Recall We recall from Lecture 8. r ( t ) = lim t r ( t + t )- r ( t ) t computes a derivative of a vector-valued function t ( ) = r ( t ) + r ( t ) ,- < < , models a tangent line to a parametric curve multiple tangent lines at the same point on a parametric curve are possible r ( t ) = ( rt- r sin( t ) ,r- r cos( t )) ,- < t < models a cycloid generated by a circle of radius r it is possible to model explicitly a tangent line to a cycloid. End of Recall 2 Arc Length . We sense that a parametric curve, when plotted on an x,y- coordinate system, has a length. We dont have a mathematical model for this; our goal is to develop such. Roughly, the idea is to partition the curve by a finite number of its points, and connect each two consecutive points by a chord; after that, sum the lengths of the chords. If the points are close enough together the sum of the chord lengths looks as if it should be a good model for our intuitive idea of arc length. We proceed with this idea. Let C denote the parametric curve modeled by the vector-valued function r ( t ) , a t b . Rather than partitioning C by pointing to the points directly, we partition C by pointing to parameter values. In that regard we let n denote a positive integer and choose n + 1 values t i [ a,b ] such that a = t < t 1 < t 2 < < t n = b . Below we denote such a partition of [ a,b ] by { t i } or by { t i } n i =0 (when n is needed in the discussion). Similarly, we write { r ( t i ) } or { r ( t i ) } n i =0 to denote the corresponding partition points on C . If the x and y components of r ( t ) are given by r ( t ) = ( x ( t ) ,y ( t )) , a t b , then for each i , the length of the chord between r ( t i- 1 ) and r ( t i ) is given by | r ( t i )- r ( t i- 1 ) | = p ( x ( t i )- x ( t i- 1 )) 2 + ( y ( t i )- y ( t i- 1 )) 2 . The sum of the chord lengths is denoted by n i =1 | r ( t i )- r ( t i- 1 ) | where it is understood that n X i =1 | r ( t i )- r ( t i- 1 ) | = | r ( t 1 )- r ( t ) | + | r ( t 2 )- r ( t 1 ) | + + | r ( t n )- r ( t n- 1 ) | . MODELING EXAMPLE.ARC LENGTH COMPUTA- TION We approximate (crudely) the arc length of the parametric curve modeled by r ( t ) = ( x ( t ) ,y ( t )) = ( t, 1- t 2 ) ,- 1 t 1 ....
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Lecture_9 - CALCULUS II Spring 2009 LECTURE 9 This lecture...

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