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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 ( θ ) + ( r ( θ )) 2 dθ • calculation of two spiral arc lengths. 1 Recall We recall from Lecture 8. • r ( t ) = lim Δ t → r ( t +Δ t ) r ( t ) Δ t computes a derivative of a vectorvalued function • t ( λ ) = r ( t ) + λ r ( t ) ,∞ < λ < ∞ , 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 don’t 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 vectorvalued 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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This note was uploaded on 02/09/2009 for the course MATH 1020 taught by Professor Ecker during the Spring '08 term at Rensselaer Polytechnic Institute.
 Spring '08
 ECKER
 Arc Length

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