of a function in a local neighborhood of a point (here this point is chosen as 0 without loss of generality) is nothing but a power series expansion! The higher he order of polynomials that our scheme can integrate the higher the order of he remainder term in the expansion. The integral of the remainer over the domain is precisely the error in numerical integration 3.3.4 Estimating the Error Using the Taylor series results and the exactness criteria ∫(x)dr-∑1tf(x) 1/3t y(2(a)2/+Idz assuming deriv (+1)! Remainder ives of f(a) are bounded on [0, 1 ()d-∑m()57+ Assume that our scheme is exact upto a polynomial order l. That means we can integrate the first (L+1) terms in this Taylor series expansion exactly. The E=/f(x)dx-∑mf(x) +1f(2(x)x1+1d +1 3.3.5 Meeting the Exactness Criteria SLIDE 12 P(x)d=/(a+a1x+a2x2+…+ad)dx=∑p(x1) Equivalently do (x) This slide needs little clarification. Our exactness criterion is ()d=/(+0+2+…+)h=∑mn()❉❏❵✿❍❝❵r❡✺❊✺❂➧●■✾❭❉❋❊✩✾❭❊✩❍❯❴❭❉❲❂✓❍✮❴✴❊✑❀✗✾❭❩❋P✙♣✭❉❋◆❖P✑❉✙❉❲❣☞❉❏❵❫❍❝▼✰❉✮✾❁❊❑●✖❡rP✺❀✓◆■❀❈●❖P✑✾❆❅➉▼✭❉❋✾❭❊❑●❬✾❆❅❬❂◗P✑❉❑❅❇❀✗❊✩❍✮❅✁￾ ❃❄✾❭●❖P✑❉❋❡❲●▲❴❭❉❑❅❖❅⑨❉✮❵✣❩❋❀✓❊✑❀✗◆■❍✮❴❭✾❭●q❛✳❢✿✾❆❅✉❊✑❉❏●■P✑✾❁❊✑❩❦♣✑❡❲●❬❍➫▼✰❉✻❃➍❀✗◆✉❅❖❀✓◆■✾❁❀✗❅✉❀✠Ñ❲▼✺❍✮❊✺❅❖✾❭❉❋❊ ✎✣➾✉P✺❀❈P✑✾❭❩❋P✑❀✓◆ ●❖P✺❀③❉✮◆◗❣❲❀✓◆❄❉✮❵✣▼✰❉✮❴❁❛✙❊✑❉✮❜❝✾❆❍❏❴❆❅✉●❖P✭❍✻●▲❉✮❡✺◆❬❅❖❂◗P✺❀✓❜❝❀③❂✓❍✮❊☞✾❁❊❑●❖❀✗❩✮◆◗❍✻●❖❀❻●■P✑❀❈P✑✾❁❩✮P✑❀✗◆❄●❖P✺❀③❉✮◆◗❣❲❀✓◆❄❉✮❵ ●❖P✺❀♦◆■❀✓❜❦❍❏✾❁❊✺❣❲❀✗◆➫●❖❀✓◆■❜ ✾❁❊ ●■P✑❀♦❀✓Ñ❲▼✺❍❏❊✺❅❖✾❁❉✮❊➣✈ ➾✉P✺❀♦✾❁❊❑●❖❀✓❩❋◆■❍✮❴➍❉✮❵▲●❖P✺❀☞◆■❀✓❜❦❍✮✾❭❊✑❀✗◆➭❉✻❨❋❀✓◆➭●❖P✑❀ ❣❲❉❋❜❝❍✮✾❭❊☞✾❆❅✉▼✑◆❖❀✵❂✠✾❆❅❇❀✗❴❭❛❯●❖P✑❀❈❀✗◆❖◆■❉✮◆⑨✾❭❊♥❊✙❡✑❜❝❀✓◆■✾❆❂✓❍❏❴✧✾❁❊❑●❖❀✗❩✮◆◗❍✻●❖✾❁❉✮❊✴✈ ✍✴→ ✍✴→✄✂ ✌✎✏✓✔✵✢✥✯➆✍✑✔✗✢✥✜✣✶✷✔✗✚✣✖❢✌❬✬✛✬✻✪✺✬ ➙✰➛❑➜➞➝✴➟✕➠✑➠ ☎❬❅❖✾❁❊✑❩➫●❖P✑❀③➾✣❍✛❛✙❴❭❉❋◆✉❅❖❀✓◆■✾❁❀✗❅⑨◆❖❀✵❅❇❡✺❴❰●◗❅❄❍❏❊✺❣♦●❖P✺❀❈❀✠Ñ✑❍✮❂✠●❖❊✑❀✵❅❖❅✉❂✠◆■✾❭●❖❀✓◆■✾❆❍ ✆ ✣ ✢ ✒ ✕✥Ï✘✗✬✫❋Ï✕❄ ✻✾✽✃✿ ✢ ✔ ✃ ✒ ✕✥Ï✭✃✹✗ ✙ ✴ ✕✫✒☛✱ ✴ ✗ ✵ ✛✏✢ ✣ ✦ ❖❬ ✢ ✒ ✕☞✬Ï ✕rÏ✘✗✺✗ ✦ Ï❖❬ ✢ Ï❖❬ ✢ ✫✮Ï ❈ ❉❊ ❋ ✝❏✟✞✟◆ ✃✽❙❏✟✠ ♠❅■❅❇❡✺❜➫✾❁❊✑❩✆❣✑❀✓◆■✾❭❨✻❍✻➹ ●❖✾❁❨✮❀✵❅✉❉❏❵ ✒ ✕rÏ✘✗▲❍❏◆■❀➉♣✰❉✮❡✑❊✭❣❲❀✗❣☞❉✮❊❣✱✲✳✦✵✴✝✶ ✡ ✡ ✡ ✡ ✡ ✛✢ ✣ ✓✒ ✕rÏ✘✗❚✫✮Ï❍❄ ✫✽ ✃✿ ✢ ✔ ✃ ✒ ✕rÏ✃ ✗ ✡ ✡ ✡ ✡ ✡☞☛ ✌ ✕✚✒☛✱✾✴ ✗ ✵ ❳❉✮❊✙❨❋❀✓◆■❩✮❀✓❊✭❂✠❀❬✾❆❅ ✄✭✖❑✬✛✒⑥❵✥❍✮❅❇●✏✎☎✎ ➡➤➢✧➥➧➦➩➨✎✍ ♠❅■❅❖❡✑❜❝❀③●❖P✭❍✻●❻❉✮❡✑◆❻❅❖❂◗P✺❀✓❜❝❀➵✾❆❅❬❀✓Ñ✑❍✮❂➧●❻❡✑▼❲●■❉♦❍✆▼✭❉❋❴❭❛✙❊✑❉❋❜➫✾❆❍❏❴✣❉✮◆◗❣❲❀✗◆ ✙ ✒q➻❁✈❻➾✉P✭❍✻●➉❜❝❀✵❍❏❊✺❅➉❃➍❀ ❂✓❍✮❊❯✾❁❊❋●■❀✓❩❋◆■❍❏●❖❀▲●■P✑❀➉Ð✺◆■❅❇●✂✕✚✒ ✱✏✴ ✗✳●■❀✓◆■❜❝❅⑨✾❁❊❯●■P✑✾❁❅✉➾✣❍✛❛✙❴❭❉❋◆⑨❅❖❀✓◆■✾❭❀✵❅✿❀✓Ñ✙▼✭❍❏❊✺❅❖✾❭❉❋❊❯❀✠Ñ✑❍❋❂➧●❖❴❁❛✮✈➃➾✉P✑❀ ❀✓◆■◆■❉✮◆⑨✾❭❊♥❊✙❡✑❜❝❀✓◆■✾❁❂✗❍❏❴✧✾❁❊❑●❖❀✓❩❋◆■❍❏●❖✾❁❉✮❊ ● ✙ ✛✢ ✣ ✒ ✕rÏ ✗✬✫❋Ï ❄ ✫✽ ✃✿ ✢ ✔ ✃ ✒ ✕rÏ✃ ✗✚✙ ✴ ✕✚✒☛✱✾✴ ✗ ✵ ✛✢ ✣ ✦ ❖❬ ✢ ✒ ✕✩✬Ï✧✕rÏ ✗❚✗ ✦ Ï❖❬ ✢ Ï❖❬ ✢ ✫❋Ï❨❫ ✍✴→ ✍✴→✑✏ ☛➶✖✙✖❑✔✗✢✸✜✞✶➤✔✵✚✞✖❢✌✌➣✍✑✘❏✔✵✜✞✖✙✏✵✏ ↕✬✛✢✥✔✗✖❑✬✻✢✥✍ ➙✰➛❑➜➞➝✴➟✕➠ ✎ ❪➃Ñ✑❍✮❂✠●❖❊✑❀✵❅❖❅❄❂✓❉✮❊✺❣✑✾❰●■✾❭❉❋❊♦◆■❀✗t❑❡✑✾❁◆❖❀✵❅ ✛✢ ✣ ✍ ❖ ✕rÏ ✗✬✫❋Ï ✙ ✛✢ ✣ ✕✫✞✣ ✱✌✞ ✢ Ï✾✱✌✞✯ Ï ✯ ✱☛✡☞✡✩✡✼✱✠✞❖ Ï❖ ✗✬✫❋Ï ✙ ✫✽ ✃✿ ✢ ✔ ✃✍ ❖ ✕✥Ï✃ ✗ ❵r❉✮◆▲❍✮❊✙❛✆❅❇❀✓●❄❉❏❵ ✒☛✱ ✴❈❂✠❉✙❀✓✒❦❂✓✾❭❀✗❊❑●■❅✑✞✣ ✦ ✞ ✢ ✦❳❫✵❫❳❫❳✦ ✞ ❖ ✌✎✤✧✦✣✢✄✑✍✑✼✸✖❑✜✹✔✵✼r✒ ✛✢ ✣ ✞ ✣ ✫✮Ï✾✱ ✛✢ ✣ ✞ ✢ Ï❵✫❋Ï ✱ ✛✢ ✣ ✞✯ Ï ✯ ✫❋Ï✾✱★✡✩✡☞✡✿✱ ✛✢ ✣ ✞ ❖ Ï❖ ✫❋Ï ✙ ✫✽ ✃✿ ✢ ✔ ✃✍ ❖ ✕✥Ï✃ ✗ ➡➤➢✧➥➧➦➩➨✰➨ ➾✉P✑✾❆❅❄❅❇❴❁✾❆❣❲❀❈❊✑❀✓❀✵❣✑❅✉❴❁✾❰●❖●❖❴❁❀③❂✠❴❆❍❏◆■✾❰Ð✭❂✗❍✻●■✾❭❉❋❊➣✈ ✓❡✑◆❄❀✠Ñ✑❍✮❂✠●❖❊✑❀✵❅❖❅✉❂✓◆❖✾❭●❖❀✗◆❖✾❁❉✮❊☞✾❁❅ ✛✢ ✣ ✍ ❖ ✕rÏ ✗✬✫❋Ï ✙ ✛✢ ✣ ✕✫✞✣ ✱✌✞ ✢ Ï✾✱✌✞✯ Ï ✯ ✱☛✡☞✡✩✡✼✱✠✞❖ Ï❖ ✗✬✫❋Ï ✙ ✫✽ ✃✿ ✢ ✔ ✃✍ ❖ ✕✥Ï✃ ✗ ✔