1-7 - January 7 2005 Today • 5.2 The Definite Integral...

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Unformatted text preview: January 7, 2005 Today • § 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 )Δ x = f ( x * 1 )Δ x + f ( x * 2 )Δ x + ··· + f ( x * n )Δ x. We also studied their limits, i.e. lim n →∞ n X i =1 f ( x * i )Δ x = lim n →∞ [ f ( x * 1 )Δ x + f ( x * 2 )Δ x ··· + f ( x * n )Δ x ] . Definition: Definite Integral Let f be a continuous function defined on the interval [ a, b ]. Divide the interval [ a, b ] into n-subintervals of equal width Δ x = b- a n . Let x = a, x n = b, x i +1 = x i + Δ x Let x * i ∈ [ 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 →∞ n X i =1 f ( x * i )Δ 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 ∑ n i =1 i = 1 + 2 + ··· + n ∑ n i =1 i = n + ( n- 1) +...
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1-7 - January 7 2005 Today • 5.2 The Definite Integral...

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