MAT371defs5

MAT371defs5 - Chapter 5 1. Riemann Integrable, Riemann...

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Unformatted text preview: Chapter 5 1. Riemann Integrable, Riemann Integral . This includes defining partition, upper Riemann sum, lower Riemann sum, upper Riemann integral, lower Riemann integral . Let f : [ a,b ] R be a bounded function. M := sup { f ( x ) | a x b } , m := inf { f ( x ) | a x b } . Let P denote the partition a = x < x 1 < x 2 < < x N = b and define x i := x i- x i- 1 . The mesh or norm of this partition is defined as | P | := max { x i | 1 i N } . We define M i := sup { f ( x ) | x i- 1 x x i } , m i := inf { f ( x ) | x i- 1 x x i } . We define the upper Riemann sum and the lower Riemann sum respectively by U ( P,f ) := N X i =1 M i x i , L ( P,f ) := N X i =1 m i x i , and the upper Riemann integral and the lower Riemann integral respectively by Z b a f ( x ) dx := inf P U ( P,f ) , Z b a f ( x ) dx := sup P L ( P,f ) . We say that f is Riemann integrable on [ a,b ] if the upper and lower Riemann integrals are equal....
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This note was uploaded on 08/12/2009 for the course MAT 371 taught by Professor Thieme during the Spring '07 term at ASU.

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MAT371defs5 - Chapter 5 1. Riemann Integrable, Riemann...

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