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Intro to Analysis in-class_Part_38

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

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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 0 , 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
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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 ]. Definition 6.1.4. The partition P 0 is a refinement of P iff P P 0 . Note that adding points to the partition has two effects: 1. the lengths of the subintervals decreases, and 2. L ( f, P ) increases and U ( f, P ) decreases.
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