Approximate Integration. Left and Right Endpoint Rules. Midpoint Rule = 2. Riemann sum (approximation to the integral) Left endpoint approximation

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M lculus II Tcqus o Igros: Approm Igro -- pr 8.7 Approm Igro M lculus II Tcqus o Igros: Approm Igro -- pr 8.7 7 L d Rg Edpo Ruls Rm sum ppromo o grl L dpo ppromo Rg dpo ppromo clculus ppls d * L d R d M lculus II Tcqus o Igros: Approm Igro -- pr 8.7 8 Mdpo Rul ] [ M d L ], [ mdpo o

Trpzodl Rul Us srg l o coc c pr o wo d pos d T L [ L ] [ L ] d T [ L ] wr / d. M lculus II Tcqus o Igros: Approm Igro -- pr 8.7 9 Empl pg d l l.97..... d T [....8 ]..9....8 d M [....7.9]....998.7.9 M lculus II Tcqus o Igros: Approm Igro -- pr 8.7 Empl pg L R M T.7..998.9.7877.877.98.977.78.88.99.9.987.887.9.97 EL E R E M E T -.88.7.9 -.88 -..7. -. -...78 -. -..97. -.. Br ppromo s crss. T rror dpo ppromos r oppos sg d dcrs cor o. T Mdpo d Trpzodl ruls r mor ccur d po ppromos. Error Mdpo d Trpzodl ruls r oppos sg dcrs cor. Mdpo Rul s rror ou l o Trpzodl rul M lculus II Tcqus o Igros: Approm Igro -- pr 8.7

M lculus II Tcqus o Igros: Approm Igro -- pr 8.7 Error Bouds d ruls, Mdpo d rpzodl rrors r d I. or Suppos K E K E E E K T M T M.7.7 or or M T E E K A D B M lculus II Tcqus o Igros: Approm Igro -- pr 8.7 Smpso s Rul Usrol o coc s o ros A B A B A B A A A A d A Bd d A d B A P, P, P, M lculus II Tcqus o Igros: Approm Igro -- pr 8.7 Smpso s Rul mus v d d / v d s wr L L L

Smpso's ruls, Error Bouds Suppos K or. I ES s rror K ES 8 M lculus II Tcqus o Igros: Approm Igro -- pr 8.7 d l l.97.... Empl pg. d S [.......7.8 ]........7.8.9 d M.98 d T.977 E M T.98.977.89.7.97 M lculus II Tcqus o Igros: Approm Igro -- pr 8.7 Empl pg How lrg sould w k ordr o gur Smpso s rulpromo or d s ccur o w.? K or or Es <. 8 > 8..8 >.7 M lculus II Tcqus o Igros: Approm Igro -- pr 8.7 7

Prcc Prolms o Pg,7-,odds,,9, M lculus II Tcqus o Igros: Approm Igro -- pr 8.7 8 Impropr Igrls M lculus II Tcqus o Igros: Approm Igro -- pr 8.8 9 Impropr Igrls T rvl o grl s. T uco s dscou rvl o grl. M lculus II Tcqus o Igros: Approm Igro -- pr 8.8

Tp I: I Irvls Empl o pg A d lm A lm d lm d A M lculus II Tcqus o Igros: Approm Igro -- pr 8.8 Do o Impropr Igrl o Tp I I d ss or vr umr, d lm d provdd s lm ss. I ss or vr umr, d d lm d provdd s lm ss. T mpropr grls d d r clld covrg lm ss d dvrg lm dos d o s. I o d d d r covrg, d d d M lculus II Tcqus o Igros: Approm Igro -- pr 8.8 d Prolm 8 o Pg 7 d d lm d d d lm lm d M lculus II Tcqus o Igros: Approm Igro -- pr 8.8

Prolm o Pg 7 r r d r r u r, r d lm r r d r dr r du dr, v r r r r r dv r d r r 9 d r r dr r r 9 9 8 9 9 9 r lm r d lm 9 9 9 M lculus II Tcqus o Igros: Approm Igro -- pr 8.8 lm 9 9 lm 9 lm lm lm 9 lm 9 9 Prolm o Pg 7 d d lm d lm d u, du d u du u d du u d lm d lm lm lm d d d s dvrg. M lculus II Tcqus o Igros: Approm Igro -- pr 8.8 M lculus II Prolm 8 o Pg 7 d u l, du d l p d lm d l d p l l l du p u p p p p d l l l l p p p p l p p p lm lm l d l p p p lm l p p p lm l lm, p > p > p p pl Tcqus o Igros: Approm Igro -- pr 8.8 7

Empl o Pg 9 d s covrg p > d dvrg p. p M lculus II Tcqus o Igros: Approm Igro -- pr 8.8 7 Do o Impropr Igrl o Tp II I couous o [, d dscouous, d lm d provdd s lm ss. I couous o, ] d dscouous, d lm d provdd s lm ss. T mpropr grls d s clld covrg lm ss d dvrg lm dos o s. I s dscou c, wr < c <, d o c d d r covrg, d c c d d d c M lculus II Tcqus o Igros: Approm Igro -- pr 8.8 8 Do o Impropr Igrl o Tp II d lm d d lm c d d d d c M lculus II Tcqus o Igros: Approm Igro -- pr 8.8 9 8

Prolm 8 o Pg 7 d d d lm d d d lm d lm d s dvrg. M lculus II Tcqus o Igros: Approm Igro -- pr 8.8 7 d Prolm o Pg 7 d d lm d s d s s s s d π lm lm s s d π s covrg. M lculus II Tcqus o Igros: Approm Igro -- pr 8.8 7 Prolm o Pg 7 d d l d l l l Is s corrc? W? M lculus II Tcqus o Igros: Approm Igro -- pr 8.8 7 9

Prolm o Pg 7 d s dvrg. d d lm d lm d d l d d l l l l d l l l lm d lm l lm d lm l l M lculus II Tcqus o Igros: Approm Igro -- pr 8.8 7 omprso Ts or Impropr Igrls I d g r couous w g or.. I d s covrg, g d s covrg.. I g d s dvrg, d s dvrg. M lculus II Tcqus o Igros: Approm Igro -- pr 8.8 7 d Sc > Bu Empl o Pg 7 d s dvrg. Hc, omprso Torm d lm lm l d lm[ l ] [ l] lm l d s dvrg. M lculus II Tcqus o Igros: Approm Igro -- pr 8.8 7

d Sc > Prolm o Pg 7 Bu d s dvrg. Hc, omprso Torm d s dvrg. M lculus II Tcqus o Igros: Approm Igro -- pr 8.8 7 Empl 9 o Pg 7 d d lm Sc lm Bu d s covrg. Hc, omprso Torm d lm[ ] [ ] lm d s covrg. M lculus II Tcqus o Igros: Approm Igro -- pr 8.8 77 Prolm o Pg 7 d Sc Bu d s covrg. Hc, omprso Torm d lm lm d d lm lm lm d s covrg. M lculus II Tcqus o Igros: Approm Igro -- pr 8.8 78

Prolm o Pg 7 d Sc Ad d s covrg. d lm lm d [ ] lm [ ] lm lm d Hc, omprso Torm d s covrg. M lculus II Tcqus o Igros: Approm Igro -- pr 8.8 79 Prolm 8 o Pg 7 d l d lm d l l u l, du d lm d lm du lm l u du p u lm lm p p p p u p >, or rm lm o s. p Hc d s covrg w > d dvrg l w. M lculus II Tcqus o Igros: Approm Igro -- pr 8.8 8 Prcc Prolms o Pg 7,9,,,7,,,9,,,7,9,9,,,,7, 9, M lculus II Tcqus o Igros: Approm Igro -- pr 8.8 8

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