This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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....
View
Full
Document
This note was uploaded on 08/12/2009 for the course MAT 371 taught by Professor Thieme during the Spring '07 term at ASU.
 Spring '07
 thieme

Click to edit the document details