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Unformatted text preview: a,b ] if and only if f + ( x ) and f( x ) are integrable on [ a,b ]. Moreover, in this case, show that Z b a f ( x ) dx = Z b a f + ( x ) dxZ b a f( x ) dx. 3) Show that if f ( x ) is contiunous on [ a,b ] with a < b , f ( x ) ≥ 0 for all x ∈ [ a,b ] and R b a f ( x ) dx = 0, then f ( x ) = 0 for all x ∈ [ a,b ]. 4) a) Assume that  f ( x )  < M for all x ∈ [ a,b ]. Show that given any ± > with a < b , if n > M ( ba ) 2 ± , then  S ( f,P n )Z b a f ( x ) dx  < ± where S ( f,P n ) is any Riemann Sum associated with the regularn partition on [ a,b ]. b) How large must n be so that  S ( f,P n )Z . 2 ex 2 dx  < 1 1000 where f ( x )ex 2 and S ( f,P n ) is any Riemann Sum associated with the regularn partition on [0 , . 2]. 2...
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This note was uploaded on 04/07/2010 for the course MATH 148 taught by Professor F.zorzitto during the Winter '08 term at Waterloo.
 Winter '08
 F.ZORZITTO
 Calculus, Riemann Sums, Limits

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