Lecture 1: Numerical Integration The Trapezoidal and Simpson s Rule

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Josephine Mason
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1 Lcur : Numrical ngraion Th Trapzoidal and Simpson s Rul A problm Th probabiliy of a normally disribud (man µ and sandard dviaion σ ) vn occurring bwn h valus a and b is B A P( a x b) d () π whr a µ b - µ A and B σ σ For a disribuion wih man µ and sandard dviaion σ wha is h probabiliy of an vn occurring in h inrval 3 x. No his will giv valus of A and B.An answr can b radily obaind from a numbr of wb calculaors as P( 3 x, µ, σ ) π d () i.., ~68% of h ara in a normal disribuion lis wihin on sandard dviaion of h man. Bu how do hs calculaors work. Th objciv in () is o calcula h ingral d s hr a known ingral for h funcion Look in vain in / x hp://n.wikipdia.org/wiki/lis_of_ingrals_of_irraional_funcions So l us rcord som valus of / in our ingral domain

2 And hn plo h funcion From h dfiniion of an ingral w know ha d s h ara undr h curv. How can w sima his. A common numrical approach is o spli h problm up ino bis a discrizaion. To sar l us choos ara srips of widh 0.. Th firs wo srips of h rquirs ara will look lik his f ( ) / f ( ) / f ( ) /

3 Each srip looks lik a rapzoid his is no sricly ru sinc h op lin is no sraigh bu curvd. Nvr-h-lss i is rasonabl o assum ha h op lin is sraigh hn h ara of srip is f ( s a r ) f ( ) srip widh avrag high A s a r And of srip f ( s a r ) f ( ) A s a r And of srip i 0 f ( s a r ( i ) ) f ( s a r Ai i ) Hnc h oal ara of h n approximaing srips is n d sar n f ( ) d [ f ( s a r ( i ) ) f ( s a r i )] (3) i whr in h cas undr considraion f ( ) s a r nd nd / s a r n s a r n (4) n 0 0. NOTE on choosing h numbr of srips n (3) and (4) will approxima any gnral dfini ingral n d f ( ) d sar Th approximaion in (3) and (4) is calld h rapzoidal rul

4 A sprad sh calculaion for our xampl problm wih n 0 srips givs d.707 -valu f() Ara Sum On noing ha by () P( a x b) 0.68 (5) π Th ral answr (o 4 placs of accuracy) is so w hav only go o wo placs of accuracy. Why would you xpc h undrsima in (5)? How can w mak h answr mor accura? (Thr ar wo answrs) W could us mor srips: Ls mak a abl of rapzoidal calculaions wih n 4,8,6,3,64,.. srips n P A lo of work bu w do g hr. Th alrnaiv is o us a br approximaion for h ara ha aks ino accoun h curvd ops. Wih rapzoids w usd a sraigh lin o us quadraic. l ( ) m c approximaion a br opion may b q a a a3

5 0.900 f ( ) / f ( ) / f ( ) / This has hr dgrs of frdom w can mak hr choics for h cofficins a L us considr h firs TWO srips A suiabl quadraic approximaion q () will b h on ha coincids wih h funcion f () a boundaris of h srips, i.., a h poins,, s a r s a r s a r This quadraic is ( q( ) s a r )( s a r ) f ( s a r ) ( s a r )( s a r ) f ( s a r ) (6) ( s a r )( s a r ) f ( s a r ) Thn for h firs wo srips ingraion undr his curv approximas h ral ingral undr h curv f (), i.., sar sar sar sar f ( ) d sar sar q( ) d 3 [ f ( ) 4 f ( ) f ( ) ] s a r s a r s a r Wih h quadraic in (6) w can ingra analyically (i aks a lo of algbra bu w do nd up wih h righ hand rsul shown. So imagin ha w spli h ara of our ingral ino an EVEN numbr of srips n

6 Thn h ara of h pair of srips bwn h i h h and i srip boundaris is sar ( i ) sar ( i ) i q( ) d s a r s a r s a r sar 3 sar i [ f ( i ) 4 f ( ( i ) ) f ( ( i ) ) ] Equaion (7) is rfrrd o as Simpson s Rul. n h sam way as h rapzoidal rul i can b usd o find h dfini ingral of a gnral funcion f () by choosing an qual numbr of srips and hn calculaing and summing h approxima aras in ach srip pair by (7). This is radily don on a spradsh for valus of n,4,8,6,.. n P Rcall P / π, and NOTE ha h Simpson s Rul rquirs lss srips convrgs o 4 poin accuracy wih 8 Srips. Th Trapzoidal rquird 64. Mak sur ha you can calcula Simposn s rul and Trapzoidal rul By hand (wih a calculaor) undr xam condiions. (7)

4.1 The Uniform Distribution Def n: A c.r.v. X has a continuous uniform distribution on [a, b] when its pdf is = 1 a x b

4.1 The Uniform Distribution Def n: A c.r.v. X has a continuous uniform distribution on [a, b] when its pdf is = 1 a x b 4. Th Uniform Disribuion Df n: A c.r.v. has a coninuous uniform disribuion on [a, b] whn is pdf is f x a x b b a Also, b + a b a µ E and V Ex4. Suppos, h lvl of unblivabiliy a any poin in a Transformrs