LOCUS Defiie egrio CONCEPT NOTES. Bsic Properies. More Properies. egrio s Limi of Sum
LOCUS Defiie egrio As eplied i he chper iled egrio Bsics, he fudmel heorem of clculus ells us h o evlue he re uder curve f from o, we firs evlue he i-derivive g of f g f d d he evlue g g. Th is, re uder he curve f() from = o = is f d g g Reders who hve eve he slighes dou regrdig he discussio ove re dvised o refer o he chper o egrio Bsics efore redig o. Defiie iegrio is o ll ou jus evluig he i-derivive d susiuig he upper d lower limis. Workig hrough his chper, ou will relise h lo of echiques eis which help us i evluig he defiie iegrl wihou resorig o he (m imes edious) process of firs deermiig he i-derivive. We will develop ll hese echiques oe oe from scrch, srig wih some eremel sic properies i Secio - Secio - BASC PROPERTES () Suppose h f() < o some iervl [, ]. The, he re uder he curve = f() from = o = will e egive i sig, i.e f d This is ovious oce ou cosider how he defiie iegrl ws rrived i he firs plce; s limi of he sum of he recgles. Thus, if f() < i some iervl he he re of he recgles i h iervl will lso e egive. This proper mes h for emple, if f() hs he followig form A A A z A Fig - he will equl f d A A A A d o A A A A.
LOCUS f we eed o evlue A A A A (he mgiude of he ouded re), we will hve o clcule f d z f d f d z f d From his, i should lso e ovious h f d () f d The re uder he curve = f() from = o = is equl i mgiude u opposie i sig o he re uder he sme curve from = o =, i.e f d f d This proper is ovious if ou cosider he Newo-Leiiz formul. f g is he i-derivive of f d is g (f), he g while () f d is g g. The re uder he curve = f() from = o = c e wrie s he sum of he re uder he curve from = o = c d from = c o =, h is c f d f d f d c Le us cosider emple of his. Le c (, ) = f() A A c Fig - is cler h he re uder he curve from = o =, A is A + A. Noe h c eed o lie ewee d for his relio o hold rue. Suppose h c >. = f() A A Fig - c
LOCUS 5 Oserve h A f d A A A c c f d f d c f d f d c Alicll, his relio c e proved esil usig he Newo Leiiz s formul. () Le f g o he iervl [, ]. The, f g d. This is ecuse he curve of f() lies ove he curve of g(), or equivlel, he curve of f g lies ove he -is for [, ] = f() This is emple where f() > g() >. f ( )d = A + A = g() A A g ( )d = A while Fig - Similrl, if f g o he iervl [, ], he f d (5) g d For he iervl [, ], suppose m < f() < M. Th is, m is lower-oud for f() while M is upper oud. The, m f d M
LOCUS 6 This is ovious oce we cosider he figure elow: M D C = f() m A X = Fig - 5 B Y = Oserve h re rec AXYB f d re rec DXYC (6) Le us cosider he iegrl of f f from o. To evlue he re uder f f, we c seprel evlue he re uder f () d he re uder f () d dd he wo res (lgericll). Thus: f f d f d f d Now cosider he iegrl of kf() from = o =. To evlue he re uder kf(), we c firs evlue he re uder f() d he mulipl i k, h is: kf d k f d (7) Cosider odd fucio f(), i. e, f f. This mes h he grph of f() is smmeric ou he origi. - + Fig - 6 From he figure, i should e ovious h f d, ecuse he re o he lef side d h o he righ lgericll dd o.
LOCUS 7 Similrl, if f() ws eve, i. e, f f - + Fig - 7 f d f d ecuse he grph is smmericl ou he -is. f ou recll he discussio i he ui o fucios, fucio c lso e eve or odd ou rirr poi =. Le us suppose h f() is odd ou =, i.e f f = The pois d - lie equidis from = eiher sides of i. Fig - 8 Suppose for emple, h we eed o clcule f d. is ovious h his will e, sice we re cosiderig equl vriio o eiher side of =, i.e. he re from = o = d he re from = o = will dd lgericll o. Similrl, if f() is eve ou =, i.e. f f he we hve, for emple Fig - 9 f d f d From his discussio, ou will ge geerl ide s o how o pproch such issues regrdig eve/odd fucios.
LOCUS 8 (8) Le us cosider fucio f() o [, ] f() f() Fig - We w o somehow defie he verge vlue h f() kes o he iervl [, ]. Wh would e pproprie w o defie such verge? Le f v e he verge vlue h we re seekig. Le i e such h i is oied some c [, ] f() f =f(c) v f() f =f(c) v c Fig - We c mesure f v sig h he re uder f() from = o = should equl he re uder he verge vlue from = o =. This seems o e he ol logicl w o defie he verge (d his is how i is cull defied!). Thus f f d v fv f d This vlue is ied for les oe c, (uder he cosri h f is coiuous, of course).
LOCUS 9 Emple Fid he re uder he curve o is eire domi, i.e., evlue d Soluio: The grph for is skeched elow: Fig - Alhough he grph eeds o ifii o oh sides, he re uder he curve will sill e fiie, s we ll ow see: d Wh do we mke of? We c ke i o me Similrl, would equl. Thus, he required re is. lim k k which is. Emple Evlue he re ouded ewee d from = o =. Soluio: The give curves re skeched i he regio of ieres elow. = = = The wo curves iersec whe =, i.e = (o he posiive side). Sice < for (, ), he grph of lies elow h of Fig -
LOCUS The required re is A d Emple Fid he me vlue of f cos o, Soluio: Le f v e he required me vlue. fv / cos / d cos d si / f v Emple Evlue d Soluio: Therefore, he give iegrl ecomes c e rewrie s d d
LOCUS Someimes, o modif iegrl, pproprie susiuio hs o e used; he sme w we did i he ui o defiie egrio. For emple, iegrls coiig he epressio ( + ) c e simplified (or modified) usig he susiuio. For evluig defiie iegrl oo, we c use he pproprie susiuio, provided we chge he limis of iegrio ccordigl lso. This will ecome cler i suseque emples. Emple 5 f / d, prove h () () Soluio: () / / d d / d / / d sec d The susiuio = c ow e used o simplif his iegrl. However, we mus chge he limis of iegrio ccordig o his susiuio: sec d d f f Thus, he modified iegrl (i erms of he ew vrile ) is: d
LOCUS () he iegrl h we re cosiderig, he limis of iegrio re o, i. e,, his iervl, <. Thus,, From proper (), we c herefore s h: or / / / d... () Usig he resul of pr () for he firs d hird erms i (), we ge our desired resul: Emple 6 For >, le l f d. Fid he fucio f f d show h f e f. e Soluio: Oserve crefull he form of he fucio f() : is i he form of iegrl (of oher fucio), wih he lower limi eig fied d he upper limi eig he vrile. As vries, f() will correspodigl vr. Oe pproch h ou migh coemple o solve his quesio is evlue he i-derivive g() of l d he evlue g g which will ecome f(). However, his will ecome uecessril cumersome (Tr i!). We c, ised, proceed s follows: / l l f f d d = + Noice h he limis of iegrio of d re differe. f he were he sme, we could hve dded d esil. So we r o mke hem he sme: i, if we le, d vries from o, will vr from o. This susiuio will herefore mke he limis of iegrio of he sme s hose of :
LOCUS d d / l d l / d / d c ow e esil dded: l d l l d l l d l d We used ised of i. This does' mke differece; is he vrile of iegrio; i c e replced wih oher vrile s log s he limis of iegrio re he sme. The fil epressio shows how simplified hs ecome. We le l z d dz d he limis of iegrio ecome o l. l z dz l Thus, f e f l e e
LOCUS Emple 7 Evlue / si cos d Soluio: The give iegrl c e modified io (esil) iegrle form epressig i i form ivolvig d sec. / si cos d / cos d / sec d The susiuio = c ow e used. sec d d d d d
LOCUS 5 Emple 8 Deermie posiive ieger 5 such h e d 6 6e Soluio: Sice will o ur o e ver lrge ieger, oe migh e emped o r ou vrious vlues of i he give relio, srig from owrds, d see which oe fis. This ril-d-error pproch migh quickl give resul i his priculr emple, u wh would we do i ws possil lrger? The geerll followed pproch i such emples, where he iegrl c e chrcerised posiive ieger (clled he order of he iegrl), is o epress i i erms of lower order iegrl. f we deoe he h order iegrl, we should r o epress i erms of k where k <. Such relio is clled recursive relio. We c he simpl use his relio repeedl o order d oi he iegrl of he e order (ised of ever ime repeig he clculio of iegrio gi) We will ow use his pproch o he curre emple: Le e d To simplif, firs of ll le d d d he limis ecome o. Thus, e d e e d We ow use iegrio prs o solve his iegrl, kig s he firs fucio: e e e d d e e d e We hve hus eslished he relio ewee d i () Now oserve h c esil e evlued:...() e e d e
LOCUS 6 Usig () repeedl, we c ow oi ll he higher order iegrls:.. 5 e. 6 6e = is herefore he posiive ieger we hd se ou o deermie. Noice he power of he recursive relio h we oied i (). Usig h relio, i ws jus mer of mior clculios o successivel deermie, d from. Wihou (), we would hve o ppl iegrio prs everime, hd we used he ril-d-error pproch. Les look oher emple of his sor. e Emple 9 Evlue e d Soluio: Noice h o mer wh e, lim e so h we will oi fiie re uder he curve. Le We ppl iegrio prs o : Thus, our recursive relio is We use his repeedl ow: e d e e d is simple o deermie:! e d e = Thus,!
LOCUS 7 Emple Prove h si d for ll si Soluio: Le Wh is? si si d. si d si si d cos d si Thus, = sisfies he sed proper. How do we pproch he geerl cse? Some reflecio o he ure of he iegrl will show ou h evluig iself would e edius. Wh we could ised do is his: We hve lred show h. f we show h, our sk would e ccomplished, sice he, d ll he higher order iegrls ecome equl o, which is ; his is wh we w o prove. si si d si si si cos d si si = Therefore, N
LOCUS 8 Emple / l d Evlue, ecuse, Soluio: Creful oservio will show h he fucio l is odd ou l l co l l Thus, s discussed i proper 7, he give iegrl will ecome. Emple d. Evlue Soluio: The fucio o e iegred s ee skeched elow i he regio of ieres: f() - Fig - This fucio is discoiuous; s discussed i proper-, we c spli he required iervl of iegrio. We will do i i such w so h i ech of he su-iervls h we oi, he fucio is coiuous d c e iegred. f, he Thus, if f d f d f d d d d f d d f d geerl, for discoiuous fucio f() whose iegrl we eed o evlue, he pproch descried ove is followed. f() is seprel iegred i su iervls where i is coiuous d he resuls so oied re dded.
LOCUS 9 TRY YOURSELF - Q. Prove h si d si Q. f is odd posiive ieger, prove h / si d si Q. Show h / cos / cos d 6 Q. Show h / d l sec 8 l z Q. 5 For >, if f dz, show h z z f f Q. 6 f cos cos d where is o-egive ieger, show h, d re i A.P. Q. 7 Show h d, is equl o. Q. 8 f = / cos d, N, show h Q. 9 Prove h Q. Prove h d si cos 5 l 6 * You mus hve used susiuios i some of he quesios ove. Thik ou he vlidi of hese susiuios. s susiuio lws vlid? Or do we eed o fulfill ceri requiremes if susiuio is o e vlid?