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 →∞ n1 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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 Spring '07
 JonathanRogawski
 Math, Calculus, Arc Length, lim, representative

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