Intro to Analysis in-class_Part_38

Intro to Analysis in-class_Part_38 - Chapter 6 Integration...

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Unformatted text preview: Chapter 6 Integration 6.1 Integrals of continuous functions 1 6.1.1 Existence of the integral Let f ( x ) be a function defined on [ a,b ]. We want to define its integral R b a f ( x ) dx . Definition 6.1.1. A partition P of the interval [ a,b ] is a finite set of points { a = x ,x 1 ,x 2 ,...,x n = b } , x i < x i +1 . Definition 6.1.2. On each subinterval [ x i ,x i +1 ], of the partition P , define M i = sup f ( x ) x i- 1 ≤ x ≤ x i m i = inf f ( x ) x i- 1 ≤ x ≤ x i U ( f,P ) = n X i =1 M i ( x i- x i- 1 ) L ( f,P ) = n X i =1 m i ( x i- x i- 1 ) U ( f,P ) is the upper sum of f on the partition P , and L ( f,P ) is the lower sum of f on the partition P . 1 May 2, 2007 82 Math 413 Integration Definition 6.1.3. If sup P L ( f,P ) = inf P U ( f,P ), then the integral of f on [ a,b ] is defined to be the common value, denoted R b a f ( x ) dx . We say f is ( Riemann-) integrable on [ a,b ] and write f ∈ R [ a,b ]....
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This note was uploaded on 11/26/2011 for the course MAT 4944 taught by Professor Wong during the Fall '11 term at FSU.

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Intro to Analysis in-class_Part_38 - Chapter 6 Integration...

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