CHAPTER 5 INTEGRATIONCHAPTER 5 INTEGRATION 5.1 ESTIMATING WITH FINITE SUMS 1. f x xabœ # Since f is...
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CHAPTER 5 INTEGRATION 5.1 ESTIMATING WITH FINITE SUMS 1. f x x ab œ # Since f is increasing on , we use left endpoints to obtain Ò!ß "Ó lower sums and right endpoints to obtain upper sums. (a) x and x i x a lower sum is ˜ œ œ œ˜ œ Ê † œ ! œ "! " " " " " # # # # # # # ) œ! " # # # i i i i !ˆ ‰ ˆ‰ Š ‹ (b) x and x i x a lower sum is ˜ œ œ œ˜ œ Ê † œ ! œ † œ "! " " " " " $ " ( ( % % % # % ) $# œ! $ # # # # # i i i i 4 4 4 4 4 !ˆ ‰ ˆ‰ ˆ‰ ˆ‰ Š ‹ (c) x and x i x an upper sum is +1 ˜ œ œ œ˜ œ Ê † œ œ "! " " " " & # # # # # # # ) œ # # # i i i i 1 2 !ˆ ‰ ˆ‰ Š ‹ (d) x and x i x an upper sum is +1 ˜ œ œ œ˜ œ Ê † œ œ † œ "! " " " " " $ " $! "& % % % # % "' $# œ" % # # # # # i i i i 4 4 4 4 4 !ˆ ‰ ˆ‰ ˆ‰ ˆ‰ ˆ ‰ Š ‹ 2. f x x ab œ $ Since f is increasing on , we use left endpoints to obtain Ò!ß "Ó lower sums and right endpoints to obtain upper sums. (a) x and x i x a lower sum is ˜ œ œ œ˜ œ Ê † œ ! œ "! " " " " " # # # # # # # "' œ! " $ $ $ i i i i !ˆ ‰ ˆ‰ Š ‹ (b) x and x i x a lower sum is ˜ œ œ œ˜ œ Ê † œ ! œ œ "! " " " " " $ $' * % % % # #&' '% œ! $ $ $ $ $ $ i i i i 4 4 4 4 4 !ˆ ‰ ˆ‰ ˆ‰ ˆ‰ Š ‹ (c) x and x i x an upper sum is +1 ˜ œ œ œ˜ œ Ê † œ œ † œ "! " " " " " * * # # # # # # # # ) "' œ $ $ $ i i i i 1 2 !ˆ ‰ ˆ‰ Š ‹ (d) x and x i x an upper sum is +1 ˜ œ œ œ˜ œ Ê † œ œœ œ "! " " " " " $ "!! #& % % % # #&' '% œ" % $ $ $ $ $ i i i i 4 4 4 4 4 !ˆ ‰ ˆ‰ ˆ‰ ˆ‰ Š ‹
CHAPTER 5 INTEGRATIONCHAPTER 5 INTEGRATION 5.1 ESTIMATING WITH FINITE SUMS 1. f x xabœ # Since f is increasing on , we use left endpoints to obtainÒ!ß "Ó lower sums and right endpoints
