d Trait 2, yinu dr a idnMan, o dayd tamura, u a zurd aM, 2 u mfi dnd durica, iou and 2aaufdru, 2 u a dou axprumlidn, avaM, 2a,, 2u ardr iMidam, 1i ays zuad fdr V, 2 200i iaim 点2m2p1,mma mt, 2M u 2a0u, d Mhk w, tr uu, 2ddM 3.2.1 In t rovisg tdh Accurat SLIDE 5 ∫(x)d Area under the Onuo as df iu prdoint uas w, d dioid, 2u in, uroaylo, 1] in, d MVin, wray 0, 0.5 and [0.5, 1] and o ri, a, 2u in,tray 0.5 f(adr=? f(a)da[ f (a)dr und appy a u idpdin, ray, d, 2u in, utraydn uai 2 NaVin, aaroay Wa d wain a hi2uu u gdo n in, 2u Midu T2u fai, dr E appuarint in frdn, df f(E) and f() aru jaM, 2u ddu ain ymt, 2M b Exercise 1 Nan Sda idu u ap o 1, 2 an pruMlidn fdr, 2u rdr in, 2iMi aM? do uai 2 ddum,2u aii arai s iu prd? P ∫(x)d♠ ❅❇●❖◆◗❍❏✾❁❩✮P❑●✿❴❁✾❭❊✺❀❬❉✮◆⑨❍③❂✠❉❋❊✺❅❇●■❍❏❊❑●❫❃⑨❉✮❡✺❴❁❣❝❩❋❀✓❊✑❀✗◆■❍❏●❖❀▲❍❈➲✗❀✓◆■❉➵❍✮❅✿●❖P✑❀✎❅❇❀✵❂✠❉✮❊✭❣❦❣❲❀✓◆■✾❭❨✻❍❏●❖✾❁❨✮❀▲❍❏❊✭❣ ●❖P✺❀✓◆■❀✠❵r❉✮◆■❀❄●■P✑❀✎❍✮♣✭❉✻❨❋❀▲❀✠Ñ❲▼✑◆■❀✗❅■❅❖✾❭❉❋❊❝●❖❀✓❴❁❴❆❅➍❡✭❅✿●■P✺❍✻●⑨●❖P✑❀✎❀✓◆■◆■❉✮◆❫✾❆❅➍✾❁❣❲❀✗❊❑●❖✾❆❂✓❍❏❴❁❴❁❛➫➲✗❀✓◆■❉③❵r❉❋◆➍♣✰❉❏●■P ●❖P✺❀✗❅❖❀❻❂✗❍✮❅❖❀✗❅✗✈ ❝▲❉✻❃❈➚✉❍❏❊✺❉❏●❖P✺❀✓◆❝✾❭❜❝▼✰❉✮◆❖●■❍✮❊❋●❯▼✺✾❭❀✵❂✠❀♥❉❏❵❻✾❭❊✑❵r❉✮◆■❜❝❍❏●❖✾❁❉✮❊✕●❖P✺❍❏●❯❃➍❀❢❉❋♣❲●■❍✮✾❭❊✕❵r◆■❉✮❜ ●■P✑✾❁❅❯❴❭✾❭●❇●■❴❭❀ ❀✠Ñ❲❀✗◆■❂✓✾❁❅❖❀❦✾❁❅➵●❖P✭❍✻●➫●❖P✑❀♦❀✓◆■◆❖❉❋◆➵❅❖❂✗❍❏❴❁❀✗❅➭❍✮❅❈●■P✑❀♥❂✠❡✑♣✰❀✆❉✮❵❄●❖P✑❀☞❣❲❉✮❜❦❍❏✾❁❊➆❴❁❀✓❊✑❩✮●❖P➣✈➤➾✉P✑✾❆❅③●■❀✓❴❁❴❁❅ ❡✺❅▲●❖P✺❍❏●▲❵r❉✮◆❬❅❖❜❦❍❏❴❁❴✴❀✗❊✑❉✮❡✑❩❋P♥✾❭❊❑●❖❀✗◆❖❨✻❍✮❴❁❅❄❉❋❡✑◆➉❅❖✾❁❜➫▼✺❴❭❀③◆■❡✑❴❁❀❈✾❁❅ ✓✁￾ ♣✺❡❲●❬❵r❉❋◆▲❴❁❍✮◆❖❩❋❀✓◆❬❣✑❉✮❜❦❍❏✾❁❊ ❴❁❀✓❊✑❩✮●❖P✺❅✉❃⑨❀❻P✺❍✛❨✮❀✎●❖❉❦❅❖❀✓❀✗➴❯♣✭❀✓●❇●■❀✓◆▲❜❝❀✠●■P✑❉❲❣✑❅✗✈ ✍✴→❆➔➣→✥↔ ✂✯➆✱✞✬✻✪☎✄➣✢✥✜✣✶➤✔✗✚✣✖✝✆❝✘✮✘❋✦✴✬✻✍✑✘✮✒ ➙✰➛❑➜➞➝✴➟✟✞ ✛✢ ✣ ✒ ✕rÏ✘✗❚✫✮Ï ✔ ✴ ✗ ✒ ✖ ✴ ❉ ✘ ✱ ✴ ✗ ✒ ✖✡✠ ❉ ✘ Normalized 1-D Problem Simple Quadrature Scheme Area under the curve is approximated by two rectangles Improving the Accuracy f x( ) x 0 1 1 4 3 4 1 2 ➡➤➢✧➥➧➦✝☛ ✓❊✑❀✿❃✉❍✛❛✎❉❏❵✑✾❁❜➫▼✺◆❖❉✻❨✙✾❁❊✑❩❄❜❦❍✛❛➉♣✰❀❫●■❉❬❣❲✾❁❨✙✾❁❣❲❀❫●❖P✑❀⑨✾❁❊❋●■❀✓◆■❨✻❍❏❴❵✱✲✳✦✵✴✝✶❑✾❭❊❑●❖❉➉❅❇❡✑♣✺✾❭❊❑●❖❀✗◆❖❨✻❍✮❴❁❅ ✱✲ ✦✺✲✳❫✥✤✶ ❍❏❊✭❣❭✱✲✳❫✥✤ ✦✵✴✝✶➣❍❏❊✺❣☞❃❄◆■✾❰●■❀✎●❖P✑❀❈✾❁❊❑●❖❀✓❩❋◆■❍✮❴ ✛✢ ✣ ✒ ✕✥Ï✘✗✬✫❋Ï ✙ ✛ ✣✌☞ ✍ ✣ ✒ ✕✥Ï✘✗✬✫❋Ï ✱ ✛✢ ✣✌☞ ✍ ✒ ✕rÏ ✗✬✫❋Ï ❍❏❊✭❣ ❍❏▼✺▼✑❴❭❛➆❍➤❜❝✾❆❣❲▼✭❉❋✾❭❊❑●❦◆■❡✑❴❭❀♦●❖❉➤●■P✑❀♥✾❭❊❑●❖❀✗❩✮◆◗❍❏❴⑨❉✮❊✕❀✗❍✮❂◗P ❅❇❡✺♣✑✾❭❊❑●■❀✓◆■❨✛❍✮❴✸✈ ➸✷❀♥❉✮♣✑●■❍❏✾❁❊✕❍ ❅■❂◗P✑❀✓❜❝❀❝❅❇P✑❉✻❃❄❊➤✾❁❊➩●■P✑❀❯❅❇❴❁✾❆❣❲❀✮✈❦➾✉P✑❀➫❵✥❍✮❂✠●❖❉❋◆ ✢✯ ❍❏▼✑▼✰❀✗❍✮◆❖✾❁❊✑❩☞✾❭❊✷❵r◆❖❉❋❊❑●❻❉✮❵ ✒ ✕ ✢✯ ✗❻❍✮❊✺❣ ✒ ✕ ❈ ✎ ✗ ❍❏◆■❀✑✏q❡✺❅q●❄●■P✑❀③❣❲❉❋❜❝❍✮✾❭❊♥❴❁❀✓❊✑❩✮●❖P✺❅✗✈ ✒✔✓✖✕➦✘✗✚✙✜✛✣✢✮➦➆➨ ❳❍✮❊➩❛❋❉✮❡➩❂✓❉✮❜❝❀➵❡✑▼✷❃❄✾❰●■P➤❍✮❊✩❀✠Ñ❲▼✑◆■❀✗❅■❅❖✾❭❉❋❊♥❵r❉✮◆✎●❖P✑❀➫❀✓◆■◆■❉✮◆❬✾❁❊✩●■P✑✾❆❅✎❂✗❍✮❅❖❀ ✛ ✦▲❉✻❃ ❜➵❡✺❂◗P♥❣❲❉✙❀✗❅⑨●■P✑❀③❍✮❂✗❂✠❡✑◆◗❍✮❂✓❛❦✾❭❜❝▼✑◆■❉✻❨✮❀ ✛ ✍✴→❆➔➣→❆➔ ❘✖✙✜✣✖❋✬✻✍✑✼✉✜✥✤✧✦➉✪✭✢✸✜✹✔✩★✞✪✺✬✻✯➤✦✞✼✸✍ ➙✰➛❑➜➞➝✴➟✟✪ ✛✢ ✣ ✒ ✕✥Ï✘✗✬✫❋Ï❂✔ ✫✽ ✃✿ ✢ ✴ ➷ ✒ ✖ ❐ ❄ ✢✯ ➷ ✘ SMA-HPC ©1999 MIT Normalized 1-D Problem Simple Quadrature Scheme f ( ) x x 0 1 ✬