1730 June 14 lecture.pdf - MAT 1730 5.3 Riemann Sums...

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1MAT 1730 5.3 Riemann Sums & Definite Integrals def: Let fbe defined on [ , ]a b, and let be a partition [ , ]a b, (not necessarily of equal lengths), given by 0121nnaxxxxxb, where ixis the width of the ithsubinterval 1[,]iixx. The Riemann sum of ffor is 11(), where .niiiiiif cxxcxdef: The norm of the partition, is the width of the largest subinterval of the partition. def: A partition is regular if baxn  . Note: In general, ban. Note: If banand 0 , then n_________. def: If fis defined on [ , ]a band 101lim()exists, where ,niiiiiif cxxcx then fis integrable on [ , ]a b, and 01lim()( )nbiiaif cxf x dx . def: ( )baf x dxthe definite integral of ffrom ato b. ais the lower limit of integration, and bis the upper limit of integration. Theorem: If a function fis continuous on [ , ]a b, then fis integrable on [ , ]a b. e.g. Write the limit as a definite integral on the given interval, if1iiixcx: a. 0

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