lect116_29rev_f11

lect116_29rev_f11 - , f ( x i 1 )) and P i = ( x i , f ( x...

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Friday, November 18 Lecture 29 : Curve length. (Refers to Section 7.1 your text) Students who have mastered the content of this lecture know : About the curve length or arclength of the curve of a function f(x). Students who have practiced the techniques presented in this lecture will be able to : Use the curve length formula to determine the length of a curve of a function f(x) over an interval. 29.1 Measuring the length of a curve Suppose we are given the curve of a function y = f ( x ) as x ranges from a to b . To measure its length we can proceed in way that is similar to what we did earlier to measure the size of the area bounded by the curve of this function. We will partition the interval [a, b] into n equal subintervals x 0 = a , x 1 , x 2 , …, x n 1 , x n = b each of length ( b a )/ n = x = x i x i 1 . For i = 1 to n let P i = ( x i , f ( x i )) and let | P i − 1 Pi | denote the length of the straight line joining the points P i − 1 = ( x i − 1
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Unformatted text preview: , f ( x i 1 )) and P i = ( x i , f ( x i )) = ( x i , y i ). We will define the length of the curve of y = f ( x ) over [ a , b ] , as Where the expression approximates the length of the curve of y = f ( x ) with a level of accuracy depending on the size of the number n . For each i, the MVT states that there exists an x i * in [ x i 1 , x i ] such that Then The last expression is a Riemann sum generated by the function [1 + f ( x ) 2 ] 1/2 . And so its limit as n tends to infinity is the definite integral of this function over [ a , b ]. So have So a formula for computing the curve length is 29.2 Example Compute the length of the arc of the parabola y = x 3/2 from (0, 0) to (1, 1). 29.3 Example Set up the integral (but do not solve) which can be used to compute the length of the portion of the curve of xy = 1 from the point (1, 1) to the point (2, )....
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lect116_29rev_f11 - , f ( x i 1 )) and P i = ( x i , f ( x...

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