The definite Riemann integral

Transcription

1 Roberto s Notes o Itegrl Clculus Chpter 4: Defte tegrls d the FTC Secto 4 The defte Rem tegrl Wht you eed to kow lredy: How to ppromte the re uder curve by usg Rem sums. Wht you c ler here: How to use the lmt process to obt the ect re uder curve. We hve see tht by usg Rem sums we c ppromte the re uder postve d cotuous fucto to y degree of ccurcy we eed. Here s remder of tht process, whe doe uder the smple ssumpto of equl wdth tervls. Strtegy to ppromte the re uder fucto f, to obt Gve postve fucto ppromte vlue for the re uder t betwee two vlues d b:. Slce the rego to th vertcl strps of wdth b.. Appromte the re of ech strp by usg rectgles of wdth d heght f, 3. where s sutbly chose vlue wth the strp 4. Add up the res of ll such rectgles to obt the Rem sum f Although ths s very good prctcl soluto to the problem, t s stll ustsfctory becuse: It s oly ppromto. It oly pples to res uder postve fucto. It requres severl choces, both for d for ech, wth ech choce provdg dfferet swer. It s dffcult to mplemet, s t requres my fucto evlutos, wth ther umber cresg f we wt better ppromtos. The otto s frly cumbersome. Itegrl Clculus Chpter 4: Defte tegrls d the FTC Secto 4: The defte Rem tegrl Pge

2 But we hve clculus tool tht wll llow us to resolve the frst ssue d, hopefully, mke progress o the others. Tht tool, of course, s lmts, d here s how ts cluso completes our strtegy. Strtegy to compute the re uder curve f, to compute the Gve postve fucto ect vlue for the re of the rego uder t betwee two vlues d b:. Slce the tervl b, to th vertcl strps of wdth b.. Appromte the re of ech strp by usg rectgles of wdth f, d heght where s vlue the strp 3. Add up the res of ll such rectgles to obt the Rem sum f 4. Compute the lmt of the Rem sum s the umber of slces pproches. If ths lmt ests d does ot deped o the choces volved, the re of the rego s defed s: A lm f As we kow, ot ll lmts est, so there re stutos where the lmt we seek my ot be there. Or t my be tht t s dfferet ccordg to how we choose the tervls d the vlues wth ech. So, we cot strt jumpg up d dow wth ectemet yet. Let me pot out, oce g, some of the remg dffcultes: We re defg the re of the rego s the lmt of the Rem sum, but does ths defto gree wth wht we usully me by re? It turs out tht t does the usul cses, but ot some speclzed stutos tht you my see lter. For ow we ll ccept tht t does Although ths method my llow us to rrve t the ect re of the rego, ts de s theoretcl d t s ot cler how esly t c be mplemeted prctce. The computtol spects of the procedure stll look very complcted d most lkely prohbtve to crry out. It s ot cler whe the vlue of the lmt depeds o the choce of, of the type of Rem sum used d o how fst we let go to. Therefore, for ow we eed to tke ths de wth gr of slt: t s good de, but ts detls d opertol steps eeds to be fleshed out stll. However, sce I kow (d probbly you kow s well) tht these detls wll be resolved, let us work o the otto ssue. Ths wll, fct, help us s we ddress the other ssues. If the lm Defto f ests d does ot deped o y of the choces mde, the such lmt s clled the Rem tegrl or defte tegrl of the fucto f betwee d b d s deoted by: l m f f ( ) d b Wht do you me by f the lmt ests? Itegrl Clculus Chpter 4: Defte tegrls d the FTC Secto 4: The defte Rem tegrl Pge

3 Emple: y,, b Ad sce we re t t, here re some further mportt cosdertos to keep md whe delg wth defte tegrls. The re uder ths curve s shded the pcture, to remd you of the slcg doe the defto of ts re. Its vlue s gve by the Rem tegrl: lm d Notce how the otto o the rght s leer d clerer th the lmt of sum otto, d eve just of the Rem sum otto. Why do we cll ths Rem tegrl or defte tegrl? Why ot just cll t tegrl? Kots o your fger A defte tegrl s umber, gve by the lmt specfed ts defto, s opposed to defte tegrl, whch s fmly of fuctos. A Rem tegrl s defed through the process we hve see here of slcg, ppromtg, ddg d computg the lmt. Other processes hve bee suggested tht led to deftos tht re equvlet to the Rem defto whe the fucto s cotuous, but ot other stutos. The coc symbol Kot o your fger tht detfes defte tegrl s deed log cptl letter S d t stds for sum, sce t represets the lmt of sum: t s Sum d the lmt mkes t log. As of ow, there s o vsble coecto betwee the defte tegrl of fucto, defed s the set of ts tdervtves, d ths defte tegrl, defed s lmt d represetg re. A mportt relto wll be dscovered lter, but we re ot there yet! Oce we eplore the coecto betwee defte d defte tegrls, t wll become cler why defte tegrls re lso represeted by the elogted S tht s the tegrl sg. There re three techcl fcts tht I c ow stte for you, eve though I wll do so wthout proof. Such proofs requre techcl detls beyod our gols d they re provded my stdrd tetbooks. The frst two re very theoretcl d re here just for completeess d to elmte ther ssue from our dscussos. I the future I shll preferetlly refer to these tegrls s defte, sce we shll ot see y other type of tegrl ths course besdes the Rem oe. But do remember tht there re other types. Itegrl Clculus Chpter 4: Defte tegrls d the FTC Secto 4: The defte Rem tegrl Pge 3

4 Techcl fcts If the lmt tht defes Rem tegrl ests, t s the sme regrdless of the choce of vlues we pck ech tervl d regrdless of how fst we let. I fct, t s ot eve ecessry for the slces to hve the sme wdth, s log s the lmt process the lrgest wdth goes to. The lst oe, however, s very prctcl, s t llows us to hve some gurtees tht these tegrls est relevt cses d motvtes us to look for better wys to compute them. cotuous. However, computg t s the lmt lm my stll be very dffcult. Thk goodess, we shll soo see how to fd t esy wy! O less mportt, but terestg ote, whe Newto d Lebz developed clculus, the de of defte tegrl ws lredy roud. It ws Newto s techer, Isc Brrow, who studed ths cocept lot d cme very close to fdg the key tht opeed up the tresures of clculus. Wht Newto d Lebz dd ws detfy the coecto betwee defte d defte tegrls, thus mkg the computto of the tegrl much eser. Ths ulocked the tremedous power tht tegrls ow hve mthemtcs s well s ll other sceces. We re gog to study the propertes of these tegrls detl lter sectos, so I wll ed ths oe by settg more termology for such detls. You re lkely lredy fmlr wth some of them, but mke sure to become totlly fmlr, sce they wll be used etesvely. If Emple: Techcl fcts f s cotuous o b,, the the b tegrl f ( ) d certly ests, lthough we my stll hve problems fdg t. d Ths tegrl, whch mesures the re uder the curve y betwee d certly ests, sce the fucto we re usg s Defto b I defte tegrl f ( ) d : The fucto Itegrl Clculus Chpter 4: Defte tegrls d the FTC Secto 4: The defte Rem tegrl Pge 4 f s clled the tegrd. The vlues d b re clled the lmts of tegrto. The symbol d s clled the dfferetl of d t s mportt d ecessry prt of the tegrl, ot to be omtted or gored, sce t correspods to the wdth of the slces.

5 Emple: I ths tegrl : d The symbol dctes the lmt of sum The fucto f s the tegrd d s the dfferetl d re the lmts of tegrto. Mke sure ot to cofuse the lmts of tegrto, whch re the boudres of the rego whose re we re computg d re very prctcl, wth the lmt eeded to defe the tegrl, whch wll soo dspper from our tteto, replced by better method. Remember tht the dfferetl ws lredy mportt prt of defte tegrl, s t detfes the vrble of tegrto d plys mportt role y chges of vrble. Now we see tht t lso hs geometrc meg: t represets the wdth of ech slce. So, do t be sloppy whe delg wth t: t deserves your respect d tteto. Lookg t the coecto betwee the two types of tegrls wll be our et step, fter short eplorto of the bsc propertes of the Rem tegrl. Summry The lmt of Rem sum, s c be used to determe the re uder curve. The more effcet otto b f d s used for such lmt d t s clled the defte or Rem tegrl of the fucto betwee the two lmts of tegrto. If tht lmt ests, t s the sme, regrdless of y choces volved. Whe the tegrd s cotuous over the tervl defed by the lmts of tegrto, the the defte tegrl ests. Commo errors to vod Become fmlr wth the otto d termology d use both correctly o cosstet bss. Itegrl Clculus Chpter 4: Defte tegrls d the FTC Secto 4: The defte Rem tegrl Pge 5

6 Lerg questos for Secto I 4-4 Revew questos:. Epl wht Rem tegrl s d how t s coected to the re problem.. Idetfy the otto d termology used for ech prt of defte tegrl. Memory questos:. Wht s the otto for the defte tegrl of y f o, b?. Whch smple codto o y f gurtees tht ts defte tegrl over b, ests? 3. Wht s dded to Rem sum to mke t to Rem tegrl? 4. Wht s the geometrcl meg of the d the otto of defte tegrl? Computto questos: For ech of the tegrls preseted questos -8: ) use your clcultor to sketch the rego whose re s represeted by the gve tegrl b) costruct the lmt formul tht defes such tegrl c) descrbe the geometrc meg of ech prt of tht formul. d 3. 3s cos d d. t d 4 4. / l d 6. e d Itegrl Clculus Chpter 4: Defte tegrls d the FTC Secto 4: The defte Rem tegrl Pge 6

7 7. s d 8. cos d Epress ech of the lmts preseted questos 9- s defte tegrl. 9. lm e /. 3 lm lm e s l.. 3 s cos lm 3 3. Determe the vlue of the tegrl 4 9 d by detfyg t s the re of smple rego d usg pproprte geometrc formule Determe the vlue of the tegrl 9 4 d by detfyg t s the re of smple rego d usg pproprte geometrc formule Use left pot estmte to ppromte the vlue of 3 e d wth =6. Theory questos:. Whe does defte tegrl represet re? 3. I Rem tegrl, s t ecessry tht ll rectgulr strps hve the sme wdth?. Why do we use the symbol to represet tegrls? Itegrl Clculus Chpter 4: Defte tegrls d the FTC Secto 4: The defte Rem tegrl Pge 7

8 4. The re of whch fmlr geometrcl shpe s represeted by tegrl of the form 8 c d? 5. Wht sgle geometrcl shpe correspods to the rego whose re s gve by 3 d? 6. I order to defe Rem tegrl, s t ecessry for the tegrd to be postve? 7. The re uder the curve tegrl d y betwee d s gve by the. Here d d re mesured uts of legth d the tegrd s qudrtc fucto. So why s the vlue of the tegrl re, mesured squre uts, sted of the cubc uts (volume) tht re suggested by the product d? 8. Why c t we ssume tht cosh d ests? Wht questos do you hve for your structor? Itegrl Clculus Chpter 4: Defte tegrls d the FTC Secto 4: The defte Rem tegrl Pge 8