weight ard'n'eval ativ pirt ts ch\\ e. That give/ 2n’ degree if freedom. Herce we m( t be able ti integrate a p lyrsmial -f degree at m i(2n-1)'. Thi imple idea give ri e ts the s called/Ga a( adrat(re cheme 2.2 Po nt-We]ht Select on Cr ter 3 SLIDE 9 Re(It h ld be exact if f(a)i a pylyesmial f(x)‘a+a1x+a2x2+ x‘p(x) Select a;’afdw;’( ch that p(a)dz:∑up(x1) frIANY ply灬mial(pt、( ard ircl( dig) 7 n Order W th 2n de] rees of freedoM, L' 2n-1 Note 8 Let pr(a) derate a plyesmial sf degree l ig the variable a(a/0. We wart ts elect the weight ard integrative pirt (ch that the form( la f(x)dx‘∑u;p(x;) exact frr all plyesmial f degree(pt\ard ircl( dig)L. Obvi( ly, with 2n degree >f freedom, the be t we ca ds i l 2n-1 .2. Why the Ex3ctness Cr ter 3? Cv ider the Taylor erie for f(a) ∫(x)f(0)+ af(0) 84f(0) x2+R+1 Rit 184+1f(Gi) where t∈[0,x Note 9 Of all f( h al The rea come frsm the tr( ct( re sf Taylsr' erie expa/ The Taylor expa i\✙ ➷➃➻✧❃➍❀✗✾❭❩❋P❋●◗❅❈❍❏❊✺❣ ✙ ➷➃➻➣❀✓❨✻❍❏❴❁❡✺❍❏●❖✾❁❉✮❊➤▼✭❉❋✾❭❊❑●■❅✎●■❉❢❂◗P✑❉✙❉❋❅❖❀✮✈❦➾✉P✺❍✻●❈❩❋✾❭❨❋❀✗❅✎❡✺❅ ✙ ✗➷➃➻➣❣❲❀✗❩✮◆■❀✓❀✵❅➉❉✮❵ ❵r◆■❀✓❀✗❣✑❉✮❜♥✈ ✦▲❀✓❊✭❂✠❀❦❃⑨❀➫❜➭❡✺❅❇●③♣✭❀❯❍✮♣✑❴❭❀➫●■❉☞✾❁❊❑●❖❀✗❩✮◆◗❍✻●❖❀❦❍☞▼✰❉✮❴❁❛❑❊✺❉✮❜❝✾❁❍✮❴➃❉❏❵✉❣❲❀✓❩❋◆❖❀✗❀❝❍❏●❈❜➫❉❑❅q● ✙ ✕ ✗➷❃❄✡✴ ✗✠➻❄✈ ➾✉P✺✾❁❅❦❅❖✾❭❜❝▼✑❴❁❀♥✾❁❣✑❀✗❍✷❩✮✾❁❨✮❀✗❅➫◆■✾❁❅❖❀♦●❖❉➤●❖P✑❀❢❅❖❉➤❂✗❍❏❴❁❴❭❀✵❣ ✖ ☎❻❍❏❡✺❅■❅❝t❋❡✭❍✮❣❲◆◗❍✻●■❡✑◆❖❀ ✖ ❅■❂◗P✑❀✓❜❝❀✮✈ ✍✴→ ✍✴→❆➔ ✦➉✪✭✢✸✜✹✔ ✤✁￾✖❑✢✸✶✭✚✹✔✄✂✴✖✙✼✥✖✙✘❏✔✵✢✥✪✰✜ ↕✬✻✢r✔✵✖❋✬✻✢✸✍ ➙✰➛❑➜➞➝✴➟✆☎ ✝▲❀✗❅❖❡✑❴❰●❬❅❖P✑❉❋❡✑❴❁❣☞♣✰❀❈❀✠Ñ✑❍✮❂✠●❄✾❰❵ ✒ ✕rÏ ✗⑨✾❁❅❄❍❝▼✰❉✮❴❁❛✙❊✑❉✮❜❝✾❆❍❏❴ ✒ ✕rÏ ✗ ✙✟✞✣ ✱✠✞ ✢ Ï✾✱✠✞✯ Ï ✯ ✱☛✡☞✡☞✡✼✱✌✞❖ Ï❖ ✙✌✍❖ ✕✥Ï✘✗ s❀✗❴❭❀✵❂➧●▲Ï✭✃❖➻ ❅❄❍❏❊✺❣ ✔ ✃❖➻ ❅❄❅❖❡✺❂◗P✆●■P✺❍✻● ✛✢ ✣ ✍ ❖ ✕✥Ï✘✗❚✫✮Ï❱✙ ✫✽ ✃✿ ✢ ✔ ✃✎✍❖ ✕rÏ✭✃▼✗ ❵r❉✮◆ ♠❝✑✏➶▼✰❉✮❴❁❛✙❊✑❉✮❜❝✾❆❍❏❴✧❡✑▼❲●■❉✰❡✥❍✮❊✺❣✆✾❁❊✺❂✓❴❭❡✺❣✑✾❭❊✑❩✮❢✓✒r❒✥❮❝❉❋◆■❣✑❀✓◆ ￾✢r✔✵✚ ✗➷ ❱✣✖❑✶✭✬✻✖✙✖❑✏❝✪✰★▲★➮✬✻✖❑✖✙❱✣✪✭✯☛✔✕✒❨✙ ✗➷❍❄ ✴ ➡➤➢✧➥➧➦✗✖ ➺➣❀✓●✘✍❖ ✕rÏ✘✗➉❣❲❀✗❊✑❉❏●■❀➭❍❦▼✰❉✮❴❁❛✙❊✑❉✮❜❝✾❆❍❏❴✣❉❏❵✿❣❲❀✓❩❋◆❖❀✗❀✙✒✿✾❁❊❢●■P✑❀➵❨✻❍❏◆■✾❆❍❏♣✑❴❁❀③Ï ❡✚✞❖✜✛✙ ✲ ❢➧✈❬➸✷❀➫❃⑨❍✮❊❑● ●❖❉❦❅❖❀✓❴❁❀✗❂✠●❄●❖P✑❀❈❃⑨❀✓✾❁❩✮P❑●■❅✉❍✮❊✺❣☞✾❭❊❑●❖❀✗❩✮◆◗❍✻●■✾❭❉❋❊✆▼✰❉✮✾❁❊❋●◗❅❄❅❇❡✭❂◗P✆●■P✺❍✻●❄●■P✑❀❻❵r❉✮◆■❜➵❡✑❴❆❍ ✛✢ ✣ ✒ ✕rÏ ✗✬✫❋Ï ✙ ✫✽ ✃✿ ✢ ✔ ✃✍ ❖ ✕rÏ✃ ✗ ✾❆❅❻❀✠Ñ✑❍✮❂✠●✎❵r❉❋◆③❍❏❴❁❴✿▼✰❉✮❴❁❛❑❊✺❉✮❜❝✾❁❍✮❴❁❅✎❉✮❵✉❣❲❀✗❩✮◆■❀✓❀➫❡✑▼❲●■❉❂❡✥❍❏❊✭❣✷✾❁❊✺❂✓❴❭❡✺❣✑✾❭❊✑❩✮❢✢✒q✈ ✓♣✙❨✙✾❭❉❋❡✺❅❇❴❁❛✮➚✴❃❄✾❰●■P ✗➷➤❣❲❀✓❩❋◆❖❀✗❀✗❅⑨❉❏❵✞❵r◆❖❀✗❀✗❣❲❉❋❜♥➚❲●❖P✑❀❈♣✰❀✗❅❇●▲❃⑨❀❻❂✗❍❏❊♥❣❲❉❝✾❁❅✣✒ ✙ ✗➷✕❄ ✴✮✈ ✍✴→ ✍✴→ ✍ ￾✚✹✒ ✔✗✚✣✖❢✌✡✌➣✍✑✘✮✔✗✜✣✖✙✏✗✏ ↕✬✻✢r✔✵✖❋✬✻✢✸✍✝ ➙✰➛❑➜➞➝✴➟✕➠✥✤ ❳❉❋❊✺❅❖✾❁❣❲❀✗◆✉●❖P✑❀③➾✣❍✛❛✙❴❭❉❋◆❄❅❇❀✗◆❖✾❁❀✗❅➍❵r❉✮◆ ✒ ✕✥Ï✘✗ ✒ ✕rÏ✘✗ ✙ ✒ ✕✒✲ ✗✙✱✧✦✒ ✕✒✲ ✗ ✦ Ï Ï ✱★✡✩✡☞✡✿✱ ✴ ✒ ✵ ✦ ❖ ✒ ✕❍✲ ✗ ✦ Ï❖ Ï❖ ✱✠✪❖❬ ✢ ✪ ❖❬ ✢ ✾❆❅⑨●❖P✑❀❈✬✻✖✙✯➆✍✑✢✸✜✞❱✣✖❑✬ ✪ ❖❬ ✢ ✙ ✴ ✕✫✒☛✱ ✴ ✗ ✵ ✦ ❖❬ ✢ ✒ ✕☞✬Ï✘✗ ✦ Ï❖ ❬ ✢ Ï❖❬ ✢ ❃❄P✑❀✗◆❖❀✭✬Ï❱✯ ✱✲ ✦❇Ï✶ ➡➤➢✧➥➧➦✯✮ ✓❵➃❍❏❴❁❴✹❵r❡✑❊✺❂➧●■✾❭❉❋❊✺❅✗➚✑❃❄P✙❛♦❍❏◆■❀✎❃➍❀❈✾❁❊❑●❖❀✓◆■❀✗❅❇●❖❀✵❣✆✾❁❊♥✾❭❊❑●■❀✓❩✮◆◗❍✻●■✾❭❊✺❩➭▼✰❉✮❴❁❛✙❊✑❉✮❜❝✾❆❍❏❴❆❅✜✛❢➾✉P✺❀❈◆❖❀✵❍✮❅❖❉✮❊ ❂✠❉❋❜❝❀✗❅❈❵r◆■❉✮❜ ●❖P✺❀♦❅❇●❖◆■❡✺❂➧●■❡✑◆❖❀✆❉❏❵▲➾✞❍✛❛✙❴❁❉✮◆✵➻ ❅➵❅❇❀✗◆❖✾❁❀✗❅❈❀✓Ñ❲▼✺❍❏❊✺❅❖✾❁❉✮❊➣✈✩➾✉P✑❀♦➾✞❍✛❛✙❴❁❉✮◆③❀✓Ñ✙▼✭❍❏❊✺❅❖✾❭❉❋❊ ✰