5.3.1-eg-1

# 5.3.1-eg-1 - Δ x = 8 n = b-1 n(since a = 1 so it must be...

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Example Consider the limit of a right Riemann sum involving n intervals of equal width, lim n →∞ n X k =1 1 q 2 + 8 k n · 8 n . Express this limit as a deﬁnite integral of the form Z b 1 f ( x ) dx by ﬁnding the value of b and the integrand f ( x ). Solution: Given f ( x ) deﬁned on [ a,b ], recall that, using n subintervals of length Δ x = b - a n , and right endpoints of intervals, c k = a + k Δ x = a + k b - a n , k = 1 , 2 ,.... then Z b a f ( x ) dx = lim n →∞ n X k =1 f ( c k x = lim n →∞ n X k =1 f a + k b - a n ! b - a n ! . It appears from the form of the given summation that
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Unformatted text preview: Δ x = 8 n = b-1 n , (since a = 1), so it must be that b = 9 . In addition, we need to ﬁnd f so that 2 + 8 k n = f ( c k ) , k = 1 , 2 ,..., where in this case, c k = 1 + k 8 n , k = 1 , 2 ,.... Thus, f ( c k ) = 2 + 8 k n = 1 + ± 1 + k 8 n ² = 1 + c k , and it then follows that f ( x ) = 1 + x....
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## This note was uploaded on 04/02/2012 for the course MTH 132 taught by Professor Kihyunhyun during the Fall '10 term at Michigan State University.

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