3.2 Simple Quadrature Scheme 4 Area under the approximated by a rectangle x Note 4 Fdr, 2u, iu u wint yu, Mhidni an, ra, dn, 2u, dpi df duoarydpint a tddd nau aiiay yaa, Int, 2u in, urtraydf a far ddu ain [0, 1 aMau u, 2a,, 2u in, strand iMa T dd, 2"fani, idn,, 2dat2 o u o iau inu, 2iM aMnau p, idn ar. FirM o u aru tdint d duouydp a naion apprdai 2 fdr d wainint a tddd apprdxiu a, idn df, 2u in, utraydf, 2iMfani, idn dn, 2iMin, aoav o 2142 o u i ary, 2u lihu pyu Aaadra, ara h2wu a' u iu pyuma, 2int o u i an dd im, d rupa u, 2u in, utrayo 1,2,2 2u, 2u prddai df, 2u in, atrand, uoayaa, aNd a, a pdin, in /idu, 2u in, uroay and, 2u yant, 2 df, 2u 21i 2 in, 2iMi af imani, S Ifo u 22ddml, 2 df iun, rdid df, win, uroaw i u a=0.5, o uav, 2u 122n u"u idpdin, Aaadra, aru d uidpdin, Aaadra, aru M2 u u rupyaiuM, 2u arua andu, 2u i arou f(a)ws a ran, ant yo 2dM2ute, iM2u fani, idn f (a)upaya, d a, 2=0.5. T2w M2u uiM acaai, o 2m f(a)iMa idnMan,. Hdo uour, o 2a, iMu/Md oida MiM 2a,, 2u m2u u iMicxai, o 2m f(a) iMa invar fani, idn df a aM uy T2u u dM d woida Mo as df Mint, 2iMiMS ruajizint, 2a, o 2am f(a)iMa Trait 2, ynu,, 2u arua andur i, iM T2iM,rapuzdid 2a y,2 aM. 2a in,tray(ian Sd La, M,rS, d dario, 2iMin a ite, s diffar an, o as IMad df, 2u in, aroay wint [0,1] o u idn /idar an in, array[o, h],h>0, davit u dru tumuray Wu u aS apnd∫(x)aa,2uim, rid df,2iMn,moay=盘 )=f(2)+△( ∫ or some∈[0, runi aindar. Lu, Min, atra, 1, 2iMhxcapan Didn doar, 2 u in, ar cay[o, h] h d2f(e) f(e)dr=h/(2)+24 dr2 Hami u, 2u uardr in, 2uuidpdin, Aaadra, aru apprdxiu a, idn iM f() h df(e)✌✉❾✸❽ ￾✉➄✂✁➐☎✄✝✆✟✞➇▲➁✡✠▲➒✑➁✣➋✑➇▲➒☛✆☞￾❄➊✺➓✌✆✍✁✎✆ ➙✰➛❑➜➞➝✴➟✑✏ ✛✢ ✣ ✓✒ ✕rÏ ✗✬✫❋Ï✕✔ ✒ ✖ ✴✗✙✘ Normalized 1-D Problem Simple Quadrature Scheme f x( ) x 0 1 1 2 Area under the curve is approximated by a rectangle ➡➤➢✧➥➧➦✕✚ ✘✑❉❋◆✿●■P✑❀▲●■✾❭❜❝❀✎♣✭❀✗✾❭❊✑❩➭❴❭❀✓●■❅⑨❂✠❉❋❊✺❂✠❀✗❊❋●■◆■❍❏●❖❀▲❉❋❊❝●❖P✑❀➉●❖❉❋▼✑✾❆❂❄❉❏❵✴❣❲❀✗❨✮❀✗❴❭❉❋▼✑✾❭❊✺❩➵❍③❩❋❉❑❉❲❣❝❊✙❡✑❜❝❀✓◆■✾❆❂✓❍❏❴ ●❖❀✵❂◗P✑❊✑✾❆t❑❡✑❀▲❵r❉❋◆➍❀✗❨✛❍✮❴❭❡✭❍✻●❖✾❁❊✑❩❈●■P✑❀✎✾❭❊❑●❖❀✗❩✮◆◗❍❏❴✭❉❏❵✴❍❈❵r❡✺❊✺❂➧●■✾❭❉❋❊✒ ✕rÏ ✗❫❉❋❊❦●❖P✺❀✎❣❲❉❋❜❦❍❏✾❁❊✰✱✲ ✦❳✴✝✶✫✈✳➸✷❀ ❍✮❅■❅❖❡✑❜❝❀➉●■P✺❍✻●❬●❖P✑❀❈✾❁❊❑●❖❀✓❩❋◆■❍✮❊✺❣♦✾❁❅▲❍✜✛❖❅❖❜➫❉✙❉✮●❖P✣✢✎❵r❡✑❊✺❂➧●■✾❭❉❋❊➣➚✑●■P✑❉✮❡✺❩✮P♥❃➍❀❈❃❄✾❁❴❁❴➣❀✠Ñ✑❍❏❜❝✾❁❊✑❀❻●❖P✺✾❁❅ ❍✮❅■❅❖❡✑❜❝▼❲●❖✾❁❉✮❊☞❴❁❍❏●❖❀✗◆✗✈✁✘✣✾❭◆◗❅❇●▲❃➍❀❈❍❏◆■❀❻❩✮❉❋✾❭❊✑❩➭●❖❉❯❣✑❀✓❨✮❀✗❴❭❉❋▼☞❍➫❊✺❍✮✾❭❨❋❀❈❍❏▼✑▼✑◆■❉❋❍❋❂◗P❯❵r❉❋◆❄❉✮♣❲●◗❍❏✾❁❊✑✾❭❊✺❩ ❍➭❩✮❉✙❉❲❣❯❍❏▼✑▼✺◆❖❉✛Ñ❲✾❁❜❝❍❏●❖✾❁❉✮❊❯❉✮❵➣●❖P✺❀✎✾❁❊❑●❖❀✗❩✮◆◗❍❏❴✰❉❏❵✴●■P✑✾❆❅❫❵r❡✺❊✺❂➧●■✾❭❉❋❊♦❉❋❊❯●❖P✺✾❁❅✉✾❁❊❑●❖❀✓◆■❨✻❍❏❴➮➚✙❃❄P✑✾❁❂◗P♦❃➍❀ ❂✓❍✮❴❭❴✹●■P✑❀ ✖✠❅❖✾❁❜➫▼✺❴❭❀❈t❑❡✺❍✮❣✑◆■❍❏●❖❡✑◆■❀✎❅■❂◗P✑❀✗❜➫❀ ✖✻✈ ➾✉P✑❀✆❅❖✾❁❜➫▼✺❴❭❀✵❅q●③●■P✑✾❁❊✑❩➩❃⑨❀✆❂✓❍✮❊✲❣❲❉✩✾❆❅❈●❖❉✷◆❖❀✗▼✑❴❆❍✮❂✠❀❝●■P✑❀❯✾❁❊❑●❖❀✓❩❋◆■❍✮❴✿❃❄✾❭●❖P✲●■P✑❀❯●■P✑❀❯▼✑◆■❉❲❣❲❡✺❂➧● ❉❏❵✣●❖P✺❀❈✾❭❊❑●❖❀✗❩✮◆◗❍❏❊✺❣➣➚❲❀✓❨✻❍❏❴❁❡✺❍✻●■❀✗❣☞❍✻●❬❍➫▼✰❉✮✾❁❊❋●▲✾❁❊✺❅❖✾❁❣✑❀❻●❖P✑❀③✾❁❊❑●❖❀✗◆❖❨✻❍❏❴➮➚✑❍❏❊✭❣✆●■P✑❀❈❴❁❀✓❊✑❩✮●❖P♥❉❏❵✣●❖P✑❀ ✾❁❊❋●■❀✓◆■❨✻❍❏❴➮➚❲❃❄P✑✾❁❂◗P♥✾❁❊☞●❖P✑✾❆❅▲❂✓❍❋❅❇❀❻✾❆❅❄❡✑❊✑✾❭●q❛✮✈❫❤✫❵✣❃⑨❀③❂◗P✑❉✙❉❋❅❖❀❬●■P✑❀③▼✰❉✮✾❁❊❑●▲❉❏❵✞❀✗❨✻❍❏❴❁❡✺❍✻●■✾❭❉❋❊♥❍❋❅⑨●❖P✑❀ ❂✠❀✗❊❑●❖◆■❉✮✾❆❣➭❉✮❵✹●❖P✑❀▲✾❁❊❑●❖❀✗◆❖❨✻❍❏❴➮➚✮✾➮✈ ❀✮✈✳Ï ✙✾✲ ❫ ✤✑➚✮❃⑨❀▲❂✓❍✮❴❭❴✺●❖P✑❀❬❅■❂◗P✑❀✗❜➫❀✙✖✠❜❝✾❁❣❲▼✰❉✮✾❁❊❑●➍t❑❡✺❍✮❣✑◆■❍❏●❖❡✑◆■❀✤✖❏✈ ♠ ❜❝✾❁❣✑▼✭❉❋✾❭❊❑●♦t❑❡✺❍❋❣❲◆■❍❏●❖❡✑◆■❀♥❅❖❂◗P✑❀✗❜❝❀☞◆■❀✓▼✺❴❁❍❋❂✠❀✗❅➫●■P✑❀✩❍❏◆■❀✗❍➤❡✑❊✭❣❲❀✓◆❦●■P✑❀✩❂✠❡✺◆❖❨❋❀ ✒ ✕✥Ï✘✗❝♣✙❛ ❍ ◆■❀✗❂➧●◗❍❏❊✑❩❋❴❭❀❄❃❄P✺❉❋❅❖❀▲P✑❀✓✾❁❩✮P❑●❫✾❆❅➃●■P✑❀▲❵r❡✺❊✺❂➧●■✾❭❉❋❊ ✒ ✕rÏ ✗➃❀✓❨✻❍✮❴❭❡✺❍❏●❖❀✵❣❦❍✻●➍Ï ✙✥✲✳❫✥✤❲✈✳➾✉P✑❀❬❅■❂◗P✑❀✗❜➫❀❬✾❁❅ ❀✠Ñ✑❍❋❂➧●✿❃❄P✑❀✗❊ ✒ ✕rÏ ✗➃✾❆❅❫❍❈❂✠❉❋❊✺❅❇●■❍❏❊❑●✵✈✧✦❬❉✻❃➍❀✗❨✮❀✓◆✵➚❏❃❄P✺❍❏●❫✾❆❅✿❴❁❀✗❅■❅✿❉✮♣✙❨✙✾❁❉✮❡✺❅➃✾❆❅➃●■P✺❍✻●❫●❖P✑❀✎❅❖❂◗P✑❀✗❜❝❀ ✾❆❅➉❀✓Ñ✑❍✮❂➧●❻❃❄P✑❀✗❊ ✒ ✕rÏ ✗➉✾❁❅❻❍♦❴❭✾❁❊✑❀✗❍✮◆❬❵r❡✺❊✺❂➧●■✾❭❉❋❊✷❉✮❵❫Ï✲❍✮❅➉❃➍❀✗❴❭❴➮✈➵➾✉P✑❀➫❜❝❉❋❅❇●✎❉❋♣❑❨✙✾❁❉✮❡✺❅➉❃✉❍✛❛☞❉✮❵ ❅❖❀✓❀✓✾❁❊✑❩➫●❖P✺✾❁❅❄✾❆❅❄♣✙❛❯◆■❀✗❍✮❴❭✾❁➲✓✾❁❊✑❩➫●❖P✺❍❏●▲❃❄P✑❀✓❊ ✒ ✕✥Ï✘✗⑨✾❆❅▲❍❝❅q●■◆■❍✮✾❭❩❋P❋●❄❴❁✾❭❊✺❀✮➚❲●■P✑❀③❍❏◆■❀✗❍➭❡✑❊✺❣❲❀✗◆▲✾❭●▲✾❁❅ ❍♦●❖◆◗❍❏▼✰❀✓➲✗❉✮✾❆❣✧✈➫➾✉P✺✾❁❅✎●■◆■❍✮▼✭❀✗➲✓❉✮✾❆❣✩P✭❍✮❅❻❀✠Ñ✑❍❋❂➧●❖❴❁❛♥●❖P✺❀❯❅❖❍✮❜➫❀❝❍❏◆■❀✗❍☞❍✮❅✎●■P✑❀❝◆❖❀✵❂➧●◗❍❏❊✑❩❋❴❭❀➫❃❄P✑✾❆❂◗P ●❖P✺✾❁❅▲❅■❂◗P✑❀✗❜➫❀❻❡✺❅❖❀✗❅⑨●■❉❦❍❏▼✑▼✑◆■❉✛Ñ❲✾❭❜❦❍❏●❖❀➉●❖P✑❀❈✾❁❊❑●❖❀✓❩❋◆■❍✮❴☎❡✥❂✓❍✮❊♥❛✮❉✮❡☞❅❇❀✗❀❈❃❄P✙❛✛ ❢✠✈ ➺➣❀✓●■❅▲●❖◆■❛♦●■❉✆❣❲❀✗◆❖✾❁❨✮❀❈●■P✑✾❆❅❬✾❁❊➩❍❯❅❖❴❭✾❁❩✮P❑●■❴❭❛♥❣❲✾❭➯✰❀✗◆❖❀✗❊❋●✎❃⑨❍✛❛❋✈❄❤✐❊✺❅❇●❖❀✗❍❋❣♥❉❏❵➃●■P✑❀➵✾❁❊❑●❖❀✓◆■❨✻❍❏❴✞♣✭❀✗✾❭❊✑❩ ✱✲✳✦❳✴❳✶➍❃⑨❀♦❂✓❉✮❊✺❅❖✾❁❣✑❀✓◆➭❍✮❊➆✾❁❊❋●■❀✓◆■❨✻❍❏❴ ✱✲✳✦✩★✶✹✦✪★✬✫ ✲❢●■❉➩♣✰❀♦❍➩♣✑✾❰●❝❜❝❉✮◆■❀❯❩✮❀✗❊✑❀✓◆◗❍❏❴➮✈✩➸➤❀♦❜❦❍✛❛ ❀✠Ñ✑❍✮▼✑❊✺❣ ✒ ✕rÏ✘✗✉❍✮♣✭❉❋❡❲●❄●❖P✺❀③❂✠❀✓❊❑●■◆❖❉❋✾❁❣✆❉❏❵✣●❖P✑✾❆❅❄✾❁❊❋●■❀✓◆■❨✻❍❏❴➮➚✮✭Ï ✙ ❮✯ ✒ ✕rÏ ✗✚✙ ✒ ✕✰✭Ï✘✗✍✱✳✲ ✕rÏ ✗ ✫✒ ✕✴✭Ï ✗ ✫❋Ï ✱ ✲ ✕✥Ï✘✗ ✯ ✗✶✵ ✫ ✯ ✒ ✕✸✷❜✗ ✫❋Ï✯ ✒✺✹✼✻✾✽✿✹✼❀❂❁ ✷✖✯❣✱✲✳✦✩★✶ ❃❄P✑❀✗◆❖❀❃✲ ✕✥Ï✘✗❂✙ Ï❅❄❆✭Ï✞✈❸➾✉P✺❀✷❴❆❍✮❅❇●♦●■❀✓◆■❜ ✾❭❊ ●❖P✺❀➤❀✓Ñ❲▼✺❍❏❊✺❅❖✾❁❉✮❊❺✾❆❅♦●❖P✑❀✲➾✣❍✛❛❑❴❁❉✮◆☞❅❇❀✗◆❖✾❁❀✗❅ ◆■❀✓❜❦❍❏✾❁❊✺❣❲❀✗◆✗✈✳➺➣❀✓●■❅❄✾❁❊❑●❖❀✓❩❋◆■❍❏●❖❀➉●❖P✑✾❆❅❄❀✓Ñ❲❍✮▼✺❍❏❊✭❅❇✾❁❉✮❊♦❉✻❨✮❀✓◆⑨●■P✑❀❈✾❭❊❑●■❀✓◆■❨✛❍✮❴✚✱✲ ✦✪★✳✶ ✛ ❮ ✣ ✒ ✕rÏ ✗✬✫❋Ï ✙❇★✒ ✕✰✭Ï✘✗✍✱ ★☛❈ ✗❊❉ ✫ ✯ ✒ ✕❋✷❜✗ ✫❋Ï✯ ✦▲❀✗❊✺❂✠❀✎●■P✑❀❈❀✓◆■◆❖❉❋◆✉✾❭❊☞●❖P✺❀❈❜➫✾❆❣❲▼✰❉✮✾❁❊❑●❬t❋❡✭❍✮❣❲◆◗❍✻●■❡✑◆❖❀❻❍❏▼✺▼✑◆❖❉✛Ñ❲✾❁❜❦❍✻●❖✾❁❉✮❊✆✾❁❅ ● ✙ ✛ ❮ ✣ ✒ ✕rÏ✘✗❚✫✮Ï❍❄❃★✒ ✕✴✭Ï ✗ ✙ ★■❈ ✗❊❉ ✫ ✯ ✒ ✕❋✷❜✗ ✫❋Ï✯ ❏