121616949-math.235 - be written as p(Δ x 2 f ± t i 2 Δ x...

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9.9 Arc Length 221 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............................................................................................................................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........................................................................................................................................ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 9.20 Approximating arc length with line segments. increases, and their lengths decrease, the sum of the lengths of the line segments will approach the true arc length; see figure 9.20 . Now we need to write a formula for the sum of the lengths of the line segments, in a form that we know becomes an integral in the limit. So we suppose we have divided the interval [ a, b ] into n subintervals as usual, each with length Δ x = ( b - a ) /n , and endpoints a = x 0 , x 1 , x 2 , . . . , x n = b . The length of a typical line segment, joinging x i to x i +1 , is p x ) 2 + ( f ( x i +1 ) - f ( x i )) 2 . By the Mean Value Theorem ( 6.26 ), there is a number t i in ( x i , x i +1 ) such that f ± ( t i x = f ( x i +1 ) - f ( x i ), so the length of the line segment can
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Unformatted text preview: be written as p (Δ x ) 2 + ( f ± ( t i )) 2 Δ x 2 = p 1 + ( f ± ( t i )) 2 Δ x. The arc length is then lim n →∞ n-1 X i =0 p 1 + ( f ± ( t i )) 2 Δ x = Z b a p 1 + ( f ± ( x )) 2 dx. Note that the sum looks a bit different than others we have encountered, because the approximation contains a t i instead of an x i . In the past we have always used left endpoints (namely, x i ) to get a representative value of f on [ x i , x i +1 ]; now we are using a different point, but the principle is the same. To summarize, to compute the length of a curve on the interval [ a, b ], we compute the integral Z b a p 1 + ( f ± ( x )) 2 dx. Unfortunately, integrals of this form are typically difficult or impossible to compute exactly, because usually none of our methods for finding antiderivatives will work. In practice this means that the integral will usually have to be approximated....
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