PRINCIPLES OF ANALYSIS
PRINCIPLES OF ANALYSIS MATH 306
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This 1 page Class Notes was uploaded by Eunice Schoen on Saturday September 26, 2015. The Class Notes belongs to MATH 306 at James Madison University taught by David Carothers in Fall. Since its upload, it has received 38 views. For similar materials see /class/214025/math-306-james-madison-university in Mathematics (M) at James Madison University.
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Date Created: 09/26/15
Suppose that f ls a functlon that ls contlnuous and nonrnegatlve at least f0 some lntelval c gt 3 b We wlll use the symbol ftdt ol ln an abbleVlated fOIm fff to denote the alea bounded by the gzaph of f and the hollzontal axls between t u and t b ftdt o f wlll be called the mesng off fmm c to b The numbels c and b ale often called the llmlts of lnteglatlon ml At thls polnt we wlll not tly to make any addltlonal llstl catlon f0 what eems llke an unusual notatlon fzdz ln paltlculal we wlll not yet olrel an explanatlon f0 what the dz ls supposed to leplesent 0 how the symbol was chosen lso no e that a mm ls Just a numbel so thele ls no paltlculal Slgnl cance to be attach d to the cholce of the Vauablet le mm would leplesent exactly the same quantlty lt ls best tdt as the 816a detelmlned by the gzaph of y t when we allow ya l blet to move fzom u o We wlll extend thls de nltlon as follows 0 a a eamp e a c If lt happens that b lt u the pzecedlrg assumed the levelse wewlll de ne 7 f F0 example 2 f by de nltlon ls the same as 712 We wlll consldel the sltuatlon ln whlch u g b ln ff as the natuzal oldel addlrg h negatlye and swltchlng the oldel If the lelatlon between a and b ls lnltlally IeVeISed fzom the natulal oldel If functlon f takes on both posltlye and negatlye values dlylde the 816a between the gzaph of f an the hollzontal axis lnto pleces ln whlch fz 2 o and f 2 lt o sepalately nd the alea of the plece above t e axls and the alea below the axis attach a negatlye Slgn to any alea below the f be the xesultlrg sum hollzontal aens and let
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