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Unformatted text preview: January 7, 2005 Today â€¢ Â§ 5.2 The Definite Integral: definition and properties. 1 When solving the area problem we en countered, Riemann sums , which are expressions of the form n X i =1 f ( x * i )Î” x = f ( x * 1 )Î” x + f ( x * 2 )Î” x + Â·Â·Â· + f ( x * n )Î” x. We also studied their limits, i.e. lim n â†’âˆž n X i =1 f ( x * i )Î” x = lim n â†’âˆž [ f ( x * 1 )Î” x + f ( x * 2 )Î” x Â·Â·Â· + f ( x * n )Î” x ] . Definition: Definite Integral Let f be a continuous function defined on the interval [ a, b ]. Divide the interval [ a, b ] into nsubintervals of equal width Î” x = b a n . Let x = a, x n = b, x i +1 = x i + Î” x Let x * i âˆˆ [ x i 1 , x i ] be sample points. The definite integral of f from a to b is Z b a f ( x ) dx = lim n â†’âˆž n X i =1 f ( x * i )Î” x. Proposition: n X i =1 i = n ( n + 1) 2 n X i =1 i 2 = n ( n + 1)(2 n + 1) 6 n X i =1 i 3 = n ( n + 1) 2 2 Proof (of the first): Note that âˆ‘ n i =1 i = 1 + 2 + Â·Â·Â· + n âˆ‘ n i =1 i = n + ( n 1) +...
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 Spring '08
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 Calculus, Riemann Sums, dx, i=1

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