Define the Lower Sum L P f and Upper Sum U P f as L P f N i 1 m i P x i x i 1 U

Define the lower sum l p f and upper sum u p f as l p

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Define the Lower Sum L P ( f ) and Upper Sum U P ( f ) , as L P ( f ) = N i = 1 m i ( P ) ( x i - x i - 1 ) U P ( f ) = N i = 1 M i ( P ) ( x i - x i - 1 ) The idea is that the “area under the graph” is underestimated by L P ( f ) and overestimated by U P ( f ) .
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Upper and Lower Sums U P ( f ) = N i = 1 M i ( P ) ( x i - x i - 1 ) L P ( f ) = N i = 1 m i ( P ) ( x i - x i - 1 ) Since m i ( P ) M i ( P ) , clearly L P ( f ) U P ( f ) for any partition P .
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Refining the sums Lemma If P Q ( Q is a refinement of P ,) then L P ( f ) L Q ( f ) U Q ( f ) U P ( f ) .
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Refining the sums Lemma If P Q ( Q is a refinement of P ,) then L P ( f ) L Q ( f ) U Q ( f ) U P ( f ) . Let’s assume that Q adds just one point y to P , and x i - 1 < y < x i . Since [ x i - 1 y ] [ x i - 1 x i ] and [ y x i ] [ x i - 1 x i ] we have M i - := sup x [ x i - 1 y ] f ( x ) sup x [ x i - 1 x i ] f ( x ) = M i ( P ) and M i + := sup x [ y x i ] f ( x ) sup x [ x i - 1 x i ] f ( x ) = M i ( P ) The only term in U P ( f ) which is modified in making U Q ( f ) is: M i ( P )( x i - x i - 1 ) = M i ( P )( y - x i - 1 ) + M i ( P )( x i - y ) M i - ( y - x i - 1 ) + M i + ( x i - y ) and so U P ( f ) U Q ( f ) . If Q contains more than one point more than P , do this repeatedly. The opposite happens when we look at the inf, L P ( f )
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Further refinements Any lower sum is smaller than any upper sum, regardless of partition: Lemma Let f be bounded and P Q be any two partitions of [ a b ] . Then L P ( f ) U Q ( f ) . Proof: Recall that P Q is a common refinement for both partitions, P P Q and Q P Q . So we apply the previous Lemma, and L P ( f ) L P Q ( f ) U P Q ( f ) U Q ( f ) Let’s optimize this. I For any fixed Q , U Q ( f ) is an upper bound for the L P ( f ) , over all partitions P . I For any fixed P , U P ( f ) is a lower bound for the L Q ( f ) , over all partitions Q . I Therefore: sup { L P ( f ) | P is a partition of [ a b ] } ≤ inf { U Q ( f ) | Q is a partition of [ a b ] }
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Riemann Integrability Definition Let f be a bounded function. f is Riemann integrable on [ a b ] if sup { L P ( f ) | P is a partition of [ a b ] } = inf { U Q ( f ) | Q is a partition of [ a b ] } . For a Riemann integrable function, we define the Riemann (definite) integral, ˆ b a f ( x ) dx = sup P L P ( f ) = inf Q U Q ( f ) Is any function not Riemann integrable?
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  • Fall '13
  • Integrals, Riemann integral, Riemann sum, Riemann

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