INTEGRATION 5.1. Estimating with Finite Sums. Chapter. Area EXAMPLE 1. Approximating Area

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1 Chpter 5 INTEGRATION OVERVIEW Oe of the gret chievemets of clssicl geometr ws to oti formuls for the res d volumes of trigles, spheres, d coes. I this chpter we stud method to clculte the res d volumes of these d other more geerl shpes. The method we develop, clled itegrtio, is tool for clcultig much more th res d volumes. The itegrl hs m pplictios i sttistics, ecoomics, the scieces, d egieerig. It llows us to clculte qutities rgig from proilities d verges to eerg cosumptio d the forces gist dm s floodgtes. The ide ehid itegrtio is tht we c effectivel compute m qutities rekig them ito smll pieces, d the summig the cotriutios from ech smll prt. We develop the theor of the itegrl i the settig of re, where it most clerl revels its ture. We egi with emples ivolvig fiite sums. These led turll to the questio of wht hppes whe more d more terms re summed. Pssig to the limit, s the umer of terms goes to ifiit, the gives itegrl. While itegrtio d differetitio re closel coected, we will ot see the roles of the derivtive d tiderivtive emerge util Sectio 5.. The ture of their coectio, cotied i the Fudmetl Theorem of Clculus, is oe of the most importt ides i clculus. 5. Estimtig with Fiite Sums.5 R.5 FIGURE 5. The re of the regio R cot e foud simple geometr formul (Emple ). This sectio shows how re, verge vlues, d the distce trveled oject over time c ll e pproimted fiite sums. Fiite sums re the sis for defiig the itegrl i Sectio 5.3. Are The re of regio with curved oudr c e pproimted summig the res of collectio of rectgles. Usig more rectgles c icrese the ccurc of the pproimtio. EXAMPE Approimtig Are Wht is the re of the shded regio R tht lies ove the -is, elow the grph of = -, d etwee the verticl lies = d =? (See Figure 5..) A rchitect might wt to kow this re to clculte the weight of custom widow with shpe descried R. Ufortutel, there is o simple geometric formul for clcultig the res of shpes hvig curved oudries like the regio R. Copright 5 Perso Eductio, Ic., pulishig s Perso Addiso-Wesle 35

2 36 Chpter 5: Itegrtio (, ), 3 (, ), 5 6, 3.5 R.5 R 3, () () FIGURE 5. () We get upper estimte of the re of R usig two rectgles cotiig R. () Four rectgles give etter upper estimte. Both estimtes overshoot the true vlue for the re. While we do ot et hve method for determiig the ect re of R, we c pproimte it i simple w. Figure 5. shows two rectgles tht together coti the regio R. Ech rectgle hs width > d the hve heights d 3>, movig from left to right. The height of ech rectgle is the mimum vlue of the fuctio ƒ, otied evlutig ƒ t the left edpoit of the suitervl of [, ] formig the se of the rectgle. The totl re of the two rectgles pproimtes the re A of the regio R, A # + 3 # = 7 8 =.875. This estimte is lrger th the true re A, sice the two rectgles coti R. We s tht.875 is upper sum ecuse it is otied tkig the height of ech rectgle s the mimum (uppermost) vlue of ƒ() for poit i the se itervl of the rectgle. I Figure 5., we improve our estimte usig four thier rectgles, ech of width >, which tke together coti the regio R. These four rectgles give the pproimtio A # # + 3 # # = 5 3 =.785, which is still greter th A sice the four rectgles coti R. Suppose isted we use four rectgles cotied iside the regio R to estimte the re, s i Figure 5.3. Ech rectgle hs width > s efore, ut the rectgles re shorter d lie etirel eeth the grph of ƒ. The fuctio ƒsd = - is decresig o [, ], so the height of ech of these rectgles is give the vlue of ƒ t the right edpoit of the suitervl formig its se. The fourth rectgle hs zero height d therefore cotriutes o re. Summig these rectgles with heights equl to the miimum vlue of ƒ() for poit i ech se suitervl, gives lower sum pproimtio to the re, A 5 6 # + 3 # # + # = 7 3 =.535. This estimte is smller th the re A sice the rectgles ll lie iside of the regio R. The true vlue of A lies somewhere etwee these lower d upper sums: A Copright 5 Perso Eductio, Ic., pulishig s Perso Addiso-Wesle

3 5. Estimtig with Fiite Sums 37, 5 6, 3, , , , , () () () () FIGURE 5. () A lower sum usig 6 rectgles of equl width = >6. () A upper sum usig 6 rectgles. FIGURE 5.3 () Rectgles cotied i R give estimte for the re tht udershoots the true vlue. () The midpoit rule uses rectgles whose height is the vlue of = ƒsd t the midpoits of their ses. B cosiderig oth lower d upper sum pproimtios we get ot ol estimtes for the re, ut lso oud o the size of the possile error i these estimtes sice the true vlue of the re lies somewhere etwee them. Here the error cot e greter th the differece =.5. Yet other estimte c e otied usig rectgles whose heights re the vlues of ƒ t the midpoits of their ses (Figure 5.3). This method of estimtio is clled the midpoit rule for pproimtig the re. The midpoit rule gives estimte tht is etwee lower sum d upper sum, ut it is ot cler whether it overestimtes or uderestimtes the true re. With four rectgles of width > s efore, the midpoit rule estimtes the re of R to e A 63 6 # # # # = 7 6 # = I ech of our computed sums, the itervl [, ] over which the fuctio ƒ is defied ws sudivided ito suitervls of equl width (lso clled legth) = s - d>, d ƒ ws evluted t poit i ech suitervl: c i the first suitervl, c i the secod suitervl, d so o. The fiite sums the ll tke the form ƒsc d + ƒsc d + ƒsc 3 d + Á + ƒsc d. B tkig more d more rectgles, with ech rectgle thier th efore, it ppers tht these fiite sums give etter d etter pproimtios to the true re of the regio R. Figure 5. shows lower sum pproimtio for the re of R usig 6 rectgles of equl width. The sum of their res is , which ppers close to the true re, ut is still smller sice the rectgles lie iside R. Figure 5. shows upper sum pproimtio usig 6 rectgles of equl width. The sum of their res is , which is somewht lrger th the true re ecuse the rectgles tke together coti R. The midpoit rule for 6 rectgles gives totl re pproimtio of , ut it is ot immeditel cler whether this estimte is lrger or smller th the true re. Copright 5 Perso Eductio, Ic., pulishig s Perso Addiso-Wesle

4 38 Chpter 5: Itegrtio TABE 5. Fiite pproimtios for the re of R Numer of suitervls ower sum Midpoit rule Upper sum Tle 5. shows the vlues of upper d lower sum pproimtios to the re of R usig up to rectgles. I Sectio 5. we will see how to get ect vlue of the res of regios such s R tkig limit s the se width of ech rectgle goes to zero d the umer of rectgles goes to ifiit. With the techiques developed there, we will e le to show tht the re of R is ectl >3. Distce Trveled Suppose we kow the velocit fuctio (t) of cr movig dow highw, without chgig directio, d wt to kow how fr it trveled etwee times t = d t =. If we lred kow tiderivtive F(t) of (t) wec fid the cr s positio fuctio s(t) settig sstd = Fstd + C. The distce trveled c the e foud clcultig the chge i positio, ssd - ssd (see Eercise 93, Sectio.8). If the velocit fuctio is determied recordig speedometer redig t vrious times o the cr, the we hve o formul from which to oti tiderivtive fuctio for velocit. So wht do we do i this situtio? Whe we do t kow tiderivtive for the velocit fuctio (t), we c pproimte the distce trveled i the followig w. Sudivide the itervl [, ] ito short time itervls o ech of which the velocit is cosidered to e firl costt. The pproimte the distce trveled o ech time suitervl with the usul distce formul distce = velocit * time d dd the results cross [, ]. Suppose the sudivided itervl looks like t t t t t t 3 t (sec) with the suitervls ll of equl legth t. Pick umer t i the first itervl. If t is so smll tht the velocit rel chges over short time itervl of durtio t, the the distce trveled i the first time itervl is out st d t. If t is umer i the secod itervl, the distce trveled i the secod time itervl is out st d t. The sum of the distces trveled over ll the time itervls is D st d t + st d t + Á + st d t, where is the totl umer of suitervls. Copright 5 Perso Eductio, Ic., pulishig s Perso Addiso-Wesle

5 5. Estimtig with Fiite Sums 39 EXAMPE Estimtig the Height of Projectile The velocit fuctio of projectile fired stright ito the ir is ƒstd = 6-9.8t m>sec. Use the summtio techique just descried to estimte how fr the projectile rises durig the first 3 sec. How close do the sums come to the ect figure of 35.9 m? Solutio We eplore the results for differet umers of itervls d differet choices of evlutio poits. Notice tht ƒ(t) is decresig, so choosig left edpoits gives upper sum estimte; choosig right edpoits gives lower sum estimte. () Three suitervls of legth, with ƒ evluted t left edpoits givig upper sum: t t t 3 3 t t With ƒ evluted t t =,, d, we hve D ƒst d t + ƒst d t + ƒst 3 d t = [6-9.8sd]sd + [6-9.8sd]sd + [6-9.8sd]sd = 5.6. () Three suitervls of legth, with ƒ evluted t right edpoits givig lower sum: t t t 3 3 t t With ƒ evluted t t =,, d 3, we hve D ƒst d t + ƒst d t + ƒst 3 d t = [6-9.8sd]sd + [6-9.8sd]sd + [6-9.8s3d]sd =.. (c) With si suitervls of legth >, we get t t t 3 t t 5 t 6 t t t 3 t t 5 t 6 t 3 3 t t t A upper sum usig left edpoits: D 3.5; lower sum usig right edpoits: D These si-itervl estimtes re somewht closer th the three-itervl estimtes. The results improve s the suitervls get shorter. As we c see i Tle 5., the left-edpoit upper sums pproch the true vlue 35.9 from ove, wheres the right-edpoit lower sums pproch it from elow. The true Copright 5 Perso Eductio, Ic., pulishig s Perso Addiso-Wesle

6 33 Chpter 5: Itegrtio TABE 5. Trvel-distce estimtes Numer of egth of ech Upper ower suitervls suitervl sum sum > > > > > > vlue lies etwee these upper d lower sums. The mgitude of the error i the closest etries is.3, smll percetge of the true vlue. Error mgitude = ƒ true vlue - clculted vlue ƒ = ƒ ƒ =.3. Error percetge = %. It would e resole to coclude from the tle s lst etries tht the projectile rose out 36 m durig its first 3 sec of flight. Displcemet Versus Distce Trveled If od with positio fuctio s(t) moves log coordite lie without chgig directio, we c clculte the totl distce it trvels from t = to t = summig the distce trveled over smll itervls, s i Emple. If the od chges directio oe or more times durig the trip, the we eed to use the od s speed ƒ std ƒ, which is the solute vlue of its velocit fuctio, (t), to fid the totl distce trveled. Usig the velocit itself, s i Emple, ol gives estimte to the od s displcemet, ssd - ssd, the differece etwee its iitil d fil positios. To see wh, prtitio the time itervl [, ] ito smll eough equl suitervls t so tht the od s velocit does ot chge ver much from time t k - to t k. The st k d gives good pproimtio of the velocit throughout the itervl. Accordigl, the chge i the od s positio coordite durig the time itervl is out st k d t. The chge is positive if st k d is positive d egtive if st k d is egtive. I either cse, the distce trveled durig the suitervl is out ƒ st k d ƒ t. The totl distce trveled is pproimtel the sum ƒ st d ƒ t + ƒ st d ƒ t + Á + ƒ st d ƒ t. Copright 5 Perso Eductio, Ic., pulishig s Perso Addiso-Wesle

7 5. Estimtig with Fiite Sums 33 c c c g() () () FIGURE 5.5 () The verge vlue of ƒsd = c o [, ] is the re of the rectgle divided -. () The verge vlue of g() o [, ] is the re eeth its grph divided -. 6 f() 3 3 FIGURE 5.6 The verge vlue of ƒsd = 3 over [, ] is 3 (Emple 3). Averge Vlue of Noegtive Fuctio The verge vlue of collectio of umers,, Á, is otied ddig them together d dividig. But wht is the verge vlue of cotiuous fuctio ƒ o itervl [, ]? Such fuctio c ssume ifiitel m vlues. For emple, the temperture t certi loctio i tow is cotiuous fuctio tht goes up d dow ech d. Wht does it me to s tht the verge temperture i the tow over the course of d is 73 degrees? Whe fuctio is costt, this questio is es to swer. A fuctio with costt vlue c o itervl [, ] hs verge vlue c. Whe c is positive, its grph over [, ] gives rectgle of height c. The verge vlue of the fuctio c the e iterpreted geometricll s the re of this rectgle divided its width - (Figure 5.5). Wht if we wt to fid the verge vlue of ocostt fuctio, such s the fuctio g i Figure 5.5? We c thik of this grph s spshot of the height of some wter tht is sloshig roud i tk, etwee eclosig wlls t = d =. As the wter moves, its height over ech poit chges, ut its verge height remis the sme. To get the verge height of the wter, we let it settle dow util it is level d its height is costt. The resultig height c equls the re uder the grph of g divided -. We re led to defie the verge vlue of oegtive fuctio o itervl [, ] to e the re uder its grph divided -. For this defiitio to e vlid, we eed precise uderstdig of wht is met the re uder grph. This will e otied i Sectio 5.3, ut for ow we look t two simple emples. EXAMPE 3 The Averge Vlue of ier Fuctio Wht is the verge vlue of the fuctio ƒsd = 3 o the itervl [, ]? Solutio The verge equls the re uder the grph divided the width of the itervl. I this cse we do ot eed fiite pproimtio to estimte the re of the regio uder the grph: trigle of height 6 d se hs re 6 (Figure 5.6). The width of the itervl is - = - =. The verge vlue of the fuctio is 6> = 3. EXAMPE The Averge Vlue of si Estimte the verge vlue of the fuctio ƒsd = si o the itervl [, p]. Solutio ookig t the grph of si etwee d p i Figure 5.7, we c see tht its verge height is somewhere etwee d. To fid the verge we eed to Copright 5 Perso Eductio, Ic., pulishig s Perso Addiso-Wesle

8 33 Chpter 5: Itegrtio f() si f() si () () FIGURE 5.7 Approimtig the re uder ƒsd = si etwee d p to compute the verge vlue of si over [, p], usig () four rectgles; () eight rectgles (Emple ). clculte the re A uder the grph d the divide this re the legth of the itervl, p - = p. We do ot hve simple w to determie the re, so we pproimte it with fiite sums. To get upper sum estimte, we dd the res of four rectgles of equl width p> tht together coti the regio eeth the grph of = si d ove the -is o [, p]. We choose the heights of the rectgles to e the lrgest vlue of si o ech suitervl. Over prticulr suitervl, this lrgest vlue m occur t the left edpoit, the right edpoit, or somewhere etwee them. We evlute si t this poit to get the height of the rectgle for upper sum. The sum of the rectgle res the estimtes the totl re (Figure 5.7): A si p # p + si p # p + si p # p + si 3p # p = # p s3.d # p.69. To estimte the verge vlue of si we divide the estimted re p d oti the pproimtio.69>p.86. If we use eight rectgles of equl width p>8 ll lig ove the grph of = si (Figure 5.7), we get the re estimte A si p 8 + si p + si 3p 8 + si p + si p + si 5p 8 + si 3p + si 7p 8 # p 8 s d # p 8 = s6.d # p Dividig this result the legth p of the itervl gives more ccurte estimte of.753 for the verge. Sice we used upper sum to pproimte the re, this estimte is still greter th the ctul verge vlue of si over [, p]. If we use more d more rectgles, with ech rectgle gettig thier d thier, we get closer d closer to the true verge vlue. Usig the techiques of Sectio 5.3, we will show tht the true verge vlue is >p.6. As efore, we could just s well hve used rectgles lig uder the grph of = si d clculted lower sum pproimtio, or we could hve used the midpoit rule. I Sectio 5.3, we will see tht it does t mtter whether our pproimtig rectgles re chose to give upper sums, lower sums, or sum i etwee. I ech cse, the pproimtios re close to the true re if ll the rectgles re sufficietl thi. Copright 5 Perso Eductio, Ic., pulishig s Perso Addiso-Wesle

9 5. Estimtig with Fiite Sums 333 Summr The re uder the grph of positive fuctio, the distce trveled movig oject tht does t chge directio, d the verge vlue of oegtive fuctio over itervl c ll e pproimted fiite sums. First we sudivide the itervl ito suitervls, tretig the pproprite fuctio ƒ s if it were costt over ech prticulr suitervl. The we multipl the width of ech suitervl the vlue of ƒ t some poit withi it, d dd these products together. If the itervl [, ] is sudivided ito suitervls of equl widths = s - d>, d if ƒsc k d is the vlue of ƒ t the chose poit c k i the kth suitervl, this process gives fiite sum of the form ƒsc d + ƒsc d + ƒsc 3 d + Á + ƒsc d. The choices for the c k could mimize or miimize the vlue of ƒ i the kth suitervl, or give some vlue i etwee. The true vlue lies somewhere etwee the pproimtios give upper sums d lower sums. The fiite sum pproimtios we looked t improved s we took more suitervls of thier width. Copright 5 Perso Eductio, Ic., pulishig s Perso Addiso-Wesle

10 5. Estimtig with Fiite Sums 333 EXERCISES 5. Are I Eercises use fiite pproimtios to estimte the re uder the grph of the fuctio usig. lower sum with two rectgles of equl width.. lower sum with four rectgles of equl width. c. upper sum with two rectgles of equl width. d. upper sum with four rectgles of equl width.. ƒsd = etwee = d =.. ƒsd = 3 etwee = d =. 3. ƒsd = > etwee = d = 5.. ƒsd = - etwee = - d =. Usig rectgles whose height is give the vlue of the fuctio t the midpoit of the rectgle s se (the midpoit rule) estimte the re uder the grphs of the followig fuctios, usig first two d the four rectgles. 5. ƒsd = etwee = d =. 6. ƒsd = 3 etwee = d =. 7. ƒsd = > etwee = d = ƒsd = - etwee = - d =. Distce 9. Distce trveled The ccompig tle shows the velocit of model tri egie movig log trck for sec. Estimte the distce trveled the egie usig suitervls of legth with. left-edpoit vlues.. right-edpoit vlues. Time Velocit Time Velocit (sec) (i. / sec) (sec) (i. / sec) Distce trveled upstrem You re sittig o the k of tidl river wtchig the icomig tide crr ottle upstrem. You record the velocit of the flow ever 5 miutes for hour, with the results show i the ccompig tle. Aout how fr upstrem did the ottle trvel durig tht hour? Fid estimte usig suitervls of legth 5 with. left-edpoit vlues.. right-edpoit vlues. Time Velocit Time Velocit (mi) (m / sec) (mi) (m / sec) Copright 5 Perso Eductio, Ic., pulishig s Perso Addiso-Wesle

11 33 Chpter 5: Itegrtio. egth of rod You d compio re out to drive twist stretch of dirt rod i cr whose speedometer works ut whose odometer (milege couter) is roke. To fid out how log this prticulr stretch of rod is, ou record the cr s velocit t -sec itervls, with the results show i the ccompig tle. Estimte the legth of the rod usig. left-edpoit vlues.. right-edpoit vlues. Velocit Velocit Time (coverted to ft/ sec) Time (coverted to ft/ sec) (sec) (3 mi/ h ft/ sec) (sec) (3 mi/ h ft/ sec) Distce from velocit dt The ccompig tle gives dt for the velocit of vitge sports cr ccelertig from to mi> h i 36 sec ( thousdths of hour). Time Velocit Time Velocit (h) (mi/ h) (h) (mi/ h) mi/hr 6. Use rectgles to estimte how fr the cr trveled durig the 36 sec it took to rech mi> h.. Roughl how m secods did it tke the cr to rech the hlfw poit? Aout how fst ws the cr goig the? Velocit d Distce 3. Free fll with ir resistce A oject is dropped stright dow from helicopter. The oject flls fster d fster ut its ccelertio (rte of chge of its velocit) decreses over time ecuse of ir resistce. The ccelertio is mesured i ft>sec d recorded ever secod fter the drop for 5 sec, s show: t Fid upper estimte for the speed whe t = 5.. Fid lower estimte for the speed whe t = 5. c. Fid upper estimte for the distce flle whe t = 3.. Distce trveled projectile A oject is shot stright upwrd from se level with iitil velocit of ft> sec.. Assumig tht grvit is the ol force ctig o the oject, give upper estimte for its velocit fter 5 sec hve elpsed. Use g = 3 ft>sec for the grvittiol ccelertio.. Fid lower estimte for the height ttied fter 5 sec. Averge Vlue of Fuctio I Eercises 5 8, use fiite sum to estimte the verge vlue of ƒ o the give itervl prtitioig the itervl ito four suitervls of equl legth d evlutig ƒ t the suitervl midpoits. 5. ƒsd = 3 o [, ] 6. ƒsd = > o [, 9] 7. ƒstd = s>d + si pt o [, ].5 8. ƒstd = - cos pt o [, ].5 si t t 8 6 cos t hours 3 t Copright 5 Perso Eductio, Ic., pulishig s Perso Addiso-Wesle

12 5. Estimtig with Fiite Sums 335 Pollutio Cotrol 9. Wter pollutio Oil is lekig out of tker dmged t se. The dmge to the tker is worseig s evideced the icresed lekge ech hour, recorded i the followig tle. Time (h) 3 ekge (gl> h) Time (h) ekge (gl> h) Give upper d lower estimte of the totl qutit of oil tht hs escped fter 5 hours.. Repet prt () for the qutit of oil tht hs escped fter 8 hours. c. The tker cotiues to lek 7 gl> h fter the first 8 hours. If the tker origill cotied 5, gl of oil, pproimtel how m more hours will elpse i the worst cse efore ll the oil hs spilled? I the est cse?. Air pollutio A power plt geertes electricit urig oil. Pollutts produced s result of the urig process re removed scruers i the smokestcks. Over time, the scruers ecome less efficiet d evetull the must e replced whe the mout of pollutio relesed eceeds govermet stdrds. Mesuremets re tke t the ed of ech moth determiig the rte t which pollutts re relesed ito the tmosphere, recorded s follows. Moth J Fe Mr Apr M Ju Pollutt Relese rte (tos> d) Moth Jul Aug Sep Oct Nov Dec Pollutt Relese rte (tos> d). Assumig 3-d moth d tht ew scruers llow ol.5 to> d relesed, give upper estimte of the totl toge of pollutts relesed the ed of Jue. Wht is lower estimte?. I the est cse, pproimtel whe will totl of 5 tos of pollutts hve ee relesed ito the tmosphere? Are of Circle. Iscrie regulr -sided polgo iside circle of rdius d compute the re of the polgo for the followig vlues of :. (squre). 8 (octgo) c. 6 d. Compre the res i prts (), (), d (c) with the re of the circle.. (Cotiutio of Eercise ). Iscrie regulr -sided polgo iside circle of rdius d compute the re of oe of the cogruet trigles formed drwig rdii to the vertices of the polgo.. Compute the limit of the re of the iscried polgo s : q. c. Repet the computtios i prts () d () for circle of rdius r. COMPUTER EXPORATIONS I Eercises 3 6, use CAS to perform the followig steps.. Plot the fuctios over the give itervl.. Sudivide the itervl ito =,, d suitervls of equl legth d evlute the fuctio t the midpoit of ech suitervl. c. Compute the verge vlue of the fuctio vlues geerted i prt (). d. Solve the equtio ƒsd = sverge vlued for usig the verge vlue clculted i prt (c) for the = prtitioig. 3. ƒsd = si o [, p]. ƒsd = si o 5. ƒsd = si o c p, p d 6. ƒsd = o c p si, p d [, p] Copright 5 Perso Eductio, Ic., pulishig s Perso Addiso-Wesle

13 5. 5. Sigm Nottio d imits of Fiite Sums 335 Sigm Nottio d imits of Fiite Sums I estimtig with fiite sums i Sectio 5., we ofte ecoutered sums with m terms (up to i Tle 5., for istce). I this sectio we itroduce ottio to write sums with lrge umer of terms. After descriig the ottio d sttig severl of its properties, we look t wht hppes to fiite sum pproimtio s the umer of terms pproches ifiit. Copright 5 Perso Eductio, Ic., pulishig s Perso Addiso-Wesle

14 336 Chpter 5: Itegrtio Fiite Sums d Sigm Nottio Sigm ottio eles us to write sum with m terms i the compct form k = Á The Greek letter (cpitl sigm, correspodig to our letter S), stds for sum. The ide of summtio k tells us where the sum egis (t the umer elow the smol) d where it eds (t the umer ove ). A letter c e used to deote the ide, ut the letters i, j, d k re customr. The summtio smol (Greek letter sigm) k k The ide k eds t k. k is formul for the kth term. The ide k strts t k. Thus we c write d = k, ƒsd + ƒsd + ƒs3d + Á + ƒsd = ƒsid. The sigm ottio used o the right side of these equtios is much more compct th the summtio epressios o the left side. i = EXAMPE Usig Sigm Nottio The sum i The sum writte out, oe The vlue sigm ottio term for ech vlue of k of the sum 5 k 3 s -d k k 5 k = k k + k k s -d sd + s -d sd + s -d 3 s3d = = = 39 The lower limit of summtio does ot hve to e ; it c e iteger. Copright 5 Perso Eductio, Ic., pulishig s Perso Addiso-Wesle

15 5. Sigm Nottio d imits of Fiite Sums 337 EXAMPE Usig Differet Ide Strtig Vlues Epress the sum i sigm ottio. Solutio The formul geertig the terms chges with the lower limit of summtio, ut the terms geerted remi the sme. It is ofte simplest to strt with k = or. Strtig with k = : = Strtig with : = 5 Strtig with k = : = 6 Strtig with k = -3: = Whe we hve sum such s we c rerrge its terms, 3 sk + k d k = k = sk + d sk - d sk - 3d k =-3 sk + 7d 3 sk + k d = s + d + s + d + s3 + 3 d = s + + 3d + s d 3 3 = k + k Regroup terms. This illustrtes geerl rule for fiite sums: s k + k d = k + Four such rules re give elow. A proof tht the re vlid c e otied usig mthemticl iductio (see Appedi ). k Alger Rules for Fiite Sums. Sum Rule: ( k + k ) = k +. Differece Rule: ( k - k ) = k - 3. Costt Multiple Rule: c k = c # k (A umer c). Costt Vlue Rule: c = # c (c is costt vlue.) k k Copright 5 Perso Eductio, Ic., pulishig s Perso Addiso-Wesle

16 338 Chpter 5: Itegrtio EXAMPE 3 Usig the Fiite Sum Alger Rules HISTORICA BIOGRAPHY Crl Friedrich Guss ( ) () () (d) s3k - k d = 3 k - s - k d = = # = (c) sk + d = k + = s + + 3d + s3 # d = 6 + = 8 k s -d # k = - # k = - k Differece Rule d Costt Multiple Rule Costt Multiple Rule Sum Rule Costt Vlue Rule Costt Vlue Rule ( > is costt) Over the ers people hve discovered vriet of formuls for the vlues of fiite sums. The most fmous of these re the formul for the sum of the first itegers (Guss m hve discovered it t ge 8) d the formuls for the sums of the squres d cues of the first itegers. EXAMPE The Sum of the First Itegers Show tht the sum of the first itegers is Solutio: The formul tells us tht the sum of the first itegers is Additio verifies this predictio: k = sds5d s + d. = =. To prove the formul i geerl, we write out the terms i the sum twice, oce forwrd d oce ckwrd Á + + s - d + s - d + Á + If we dd the two terms i the first colum we get + = +. Similrl, if we dd the two terms i the secod colum we get + s - d = +. The two terms i colum sum to +. Whe we dd the colums together we get terms, ech equl to +, for totl of s + d. Sice this is twice the desired qutit, the sum of the first itegers is sds + d>. Formuls for the sums of the squres d cues of the first itegers re proved usig mthemticl iductio (see Appedi ). We stte them here. The first squres: k = The first cues: k 3 s + d = s + ds + d 6 Copright 5 Perso Eductio, Ic., pulishig s Perso Addiso-Wesle

17 5. Sigm Nottio d imits of Fiite Sums 339 imits of Fiite Sums The fiite sum pproimtios we cosidered i Sectio 5. got more ccurte s the umer of terms icresed d the suitervl widths (legths) ecme thier. The et emple shows how to clculte limitig vlue s the widths of the suitervls go to zero d their umer grows to ifiit. EXAMPE 5 The imit of Fiite Approimtios to Are Fid the limitig vlue of lower sum pproimtios to the re of the regio R elow the grph of = - d ove the itervl [, ] o the -is usig equl width rectgles whose widths pproch zero d whose umer pproches ifiit. (See Figure 5..) Solutio We compute lower sum pproimtio usig rectgles of equl width = s - d>, d the we see wht hppes s : q. We strt sudividig [, ] ito equl width suitervls Ech suitervl hs width >. The fuctio - is decresig o [, ], d its smllest vlue i suitervl occurs t the suitervl s right edpoit. So lower sum is costructed with rectgles whose height over the suitervl [sk - d>, k>] is ƒsk>d = - sk>d, givig the sum We write this i sigm ottio d simplif, c, d, c, d, Á, c -, d. ƒ + ƒ + Á + ƒ k + Á + ƒ. ƒ k = - k = - k 3 = = # - 3 k = - sds + ds + d 3 6 = k 3 Differece Rule Sum of the First Squres Numertor epded We hve otied epressio for the lower sum tht holds for. Tkig the limit of this epressio s : q, we see tht the lower sums coverge s the umer of suitervls icreses d the suitervl widths pproch zero: lim : q 6 3 = - 6 = 3. Costt Vlue d Costt Multiple Rules The lower sum pproimtios coverge to >3. A similr clcultio shows tht the upper sum pproimtios lso coverge to >3 (Eercise 35). A fiite sum pproimtio, i the sese of our summr t the ed of Sectio 5., lso coverges to the sme vlue Copright 5 Perso Eductio, Ic., pulishig s Perso Addiso-Wesle

18 3 Chpter 5: Itegrtio f() FIGURE 5.8 A tpicl cotiuous fuctio = ƒsd over closed itervl [, ]. >3. This is ecuse it is possile to show tht fiite sum pproimtio is trpped etwee the lower d upper sum pproimtios. For this reso we re led to defie the re of the regio R s this limitig vlue. I Sectio 5.3 we stud the limits of such fiite pproimtios i their more geerl settig. HISTORICA BIOGRAPHY Georg Friedrich Berhrd Riem (86 866) Riem Sums The theor of limits of fiite pproimtios ws mde precise the Germ mthemtici Berhrd Riem. We ow itroduce the otio of Riem sum, which uderlies the theor of the defiite itegrl studied i the et sectio. We egi with ritrr fuctio ƒ defied o closed itervl [, ]. ike the fuctio pictured i Figure 5.8, ƒ m hve egtive s well s positive vlues. We sudivide the itervl [, ] ito suitervls, ot ecessril of equl widths (or legths), d form sums i the sme w s for the fiite pproimtios i Sectio 5.. To do so, we choose - poits 5,, 3, Á, - 6 etwee d d stisfig To mke the ottio cosistet, we deote d, so tht The set Á 6-6. = Á 6-6 =. P = 5,,, Á, -, 6 is clled prtitio of [, ]. The prtitio P divides [, ] ito closed suitervls [, ], [, ], Á, [ -, ]. The first of these suitervls is [, ], the secod is [, ], d the kth suitervl of P is [ k -, k ], for k iteger etwee d. Copright 5 Perso Eductio, Ic., pulishig s Perso Addiso-Wesle

19 5. Sigm Nottio d imits of Fiite Sums 3 kth suitervl k k The width of the first suitervl [, ] is deoted, the width of the secod [, ] is deoted, d the width of the kth suitervl is k = k - k -. If ll suitervls hve equl width, the the commo width is equl to s - d>. k k k I ech suitervl we select some poit. The poit chose i the kth suitervl [ k -, k ] is clled c k. The o ech suitervl we std verticl rectgle tht stretches from the -is to touch the curve t sc k, ƒsc k dd. These rectgles c e ove or elow the -is, depedig o whether ƒsc k d is positive or egtive, or o it if ƒsc k d = (Figure 5.9). O ech suitervl we form the product ƒsc k d # k. This product is positive, egtive or zero, depedig o the sig of ƒsc k d. Whe ƒsc k d 7, the product ƒsc k d # k is the re of rectgle with height ƒsc k d d width k. Whe ƒsc k d 6, the product ƒsc k d # k is egtive umer, the egtive of the re of rectgle of width k tht drops from the -is to the egtive umer ƒsc k d. Fill we sum ll these products to get S P = ƒsc k d k. f() (c, f(c )) (c k, f(c k )) kth rectgle c c c k c k k (c, f(c )) (c, f(c )) FIGURE 5.9 The rectgles pproimte the regio etwee the grph of the fuctio = ƒsd d the -is. Copright 5 Perso Eductio, Ic., pulishig s Perso Addiso-Wesle

20 3 Chpter 5: Itegrtio f() S P The sum is clled Riem sum for ƒ o the itervl [, ]. There re m such sums, depedig o the prtitio P we choose, d the choices of the poits c k i the suitervls. I Emple 5, where the suitervls ll hd equl widths = >, we could mke them thier simpl icresig their umer. Whe prtitio hs suitervls of vrig widths, we c esure the re ll thi cotrollig the width of widest (logest) suitervl. We defie the orm of prtitio P, writte 7P7, to e the lrgest of ll the suitervl widths. If 7P7 is smll umer, the ll of the suitervls i the prtitio P hve smll width. et s look t emple of these ides. () EXAMPE 6 Prtitioig Closed Itervl The set P = {,.,.6,,.5, } is prtitio of [, ]. There re five suitervls of P: [,.], [.,.6], [.6, ], [,.5], d [.5, ]: f() () FIGURE 5. The curve of Figure 5.9 with rectgles from fier prtitios of [, ]. Fier prtitios crete collectios of rectgles with thier ses tht pproimte the regio etwee the grph of ƒ d the -is with icresig ccurc The legths of the suitervls re =., =., 3 =., =.5, d 5 =.5. The logest suitervl legth is.5, so the orm of the prtitio is 7P7 =.5. I this emple, there re two suitervls of this legth. A Riem sum ssocited with prtitio of closed itervl [, ] defies rectgles tht pproimte the regio etwee the grph of cotiuous fuctio ƒ d the -is. Prtitios with orm pprochig zero led to collectios of rectgles tht pproimte this regio with icresig ccurc, s suggested Figure 5.. We will see i the et sectio tht if the fuctio ƒ is cotiuous over the closed itervl [, ], the o mtter how we choose the prtitio P d the poits c k i its suitervls to costruct Riem sum, sigle limitig vlue is pproched s the suitervl widths, cotrolled the orm of the prtitio, pproch zero. Copright 5 Perso Eductio, Ic., pulishig s Perso Addiso-Wesle

21 3 Chpter 5: Itegrtio EXERCISES 5. Sigm Nottio Write the sums i Eercises 6 without sigm ottio. The evlute them. 6k.. k + 3. cos kp d 6. s k + si p k 7. Which of the followig epress i sigm ottio? 6 k - 5 k k = 5 si kp s -d k cos kp.. c. 3 k - k k =- k + 8. Which of the followig epress i sigm ottio? 6. s -d k -. s -d k k c. 9. Which formul is ot equivlet to the other two?.. c. k =. Which formul is ot equivlet to the other two?. sk - d. sk + d c. s -d k - k - 5 k = k = 3 k =- s -d k k + 3 s -d k + k + k =- k =- - k =-3 k s -d k k + Copright 5 Perso Eductio, Ic., pulishig s Perso Addiso-Wesle

22 5. Sigm Nottio d imits of Fiite Sums 33 Epress the sums i Eercises 6 i sigm ottio. The form of our swer will deped o our choice of the lower limit of summtio Vlues of Fiite Sums 7. Suppose tht k = -5 d k = 6. Fid the vlues of. 3 k. c. s k + k d 6 d. s k - k d e. s k - k d 8. Suppose tht k = d. Fid the vlues of. 8 k. 5 k c. s k + d d. s k - d Evlute the sums i Eercises k. c... k. c. 7. s -kd s3 - k d. sk - 5d 3 k k 3 k 5 pk 5 k 3 3 k 3 Rectgles for Riem Sums I Eercises 9 3, grph ech fuctio ƒ() over the give itervl. Prtitio the itervl ito four suitervls of equl legth. The dd to our sketch the rectgles ssocited with the Riem sum ƒsc k d k, give tht c k is the () left-hd edpoit, () righthd edpoit, (c) midpoit of the kth suitervl. (Mke seprte sketch for ech set of rectgles.) ks3k + 5d 6. ksk + d 5 7 k k ƒsd = -, [, ] ƒsd = -, [, ] ƒsd = si, [-p, p] ƒsd = si +, [-p, p] 33. Fid the orm of the prtitio P = 5,.,.5,.3,.6, Fid the orm of the prtitio P = 5-, -.6, -.5,,.8, 6. imits of Upper Sums For the fuctios i Eercises 35 fid formul for the upper sum otied dividig the itervl [, ] ito equl suitervls. The tke limit of these sums s : q to clculte the re uder the curve over [, ]. 35. ƒsd = - over the itervl [, ]. 36. ƒsd = over the itervl [, 3]. 37. ƒsd = + over the itervl [, 3]. 38. ƒsd = 3 over the itervl [, ]. 39. ƒsd = + over the itervl [, ].. ƒsd = 3 + over the itervl [, ]. 7 k - 7 k 3 Copright 5 Perso Eductio, Ic., pulishig s Perso Addiso-Wesle

23 The Defiite Itegrl 33 The Defiite Itegrl I Sectio 5. we ivestigted the limit of fiite sum for fuctio defied over closed itervl [, ] usig suitervls of equl width (or legth), s - d>. I this sectio we cosider the limit of more geerl Riem sums s the orm of the prtitios of [, ] pproches zero. For geerl Riem sums the suitervls of the prtitios eed ot hve equl widths. The limitig process the leds to the defiitio of the defiite itegrl of fuctio over closed itervl [, ]. imits of Riem Sums The defiitio of the defiite itegrl is sed o the ide tht for certi fuctios, s the orm of the prtitios of [, ] pproches zero, the vlues of the correspodig Riem Copright 5 Perso Eductio, Ic., pulishig s Perso Addiso-Wesle

24 3 Chpter 5: Itegrtio sums pproch limitig vlue I. Wht we me this covergig ide is tht Riem sum will e close to the umer I provided tht the orm of its prtitio is sufficietl smll (so tht ll of its suitervls hve thi eough widths). We itroduce the smol P s smll positive umer tht specifies how close to I the Riem sum must e, d the smol d s secod smll positive umer tht specifies how smll the orm of prtitio must e i order for tht to hppe. Here is precise formultio. DEFINITION The Defiite Itegrl s imit of Riem Sums et ƒ() e fuctio defied o closed itervl [, ]. We s tht umer I is the defiite itegrl of ƒ over [, ] d tht I is the limit of the Riem sums g ƒsc k d k if the followig coditio is stisfied: Give umer P7there is correspodig umer d 7 such tht for ever prtitio P = 5,, Á, 6 of [, ] with 7P7 6 d d choice of i [ k -, k ], we hve c k ` ƒsc k d k - I ` 6P. eiiz itroduced ottio for the defiite itegrl tht cptures its costructio s limit of Riem sums. He evisioed the fiite sums g ƒsc k d k ecomig ifiite sum of fuctio vlues ƒ() multiplied ifiitesiml suitervl widths d. The sum smol is replced i the limit the itegrl smol, whose origi is i the letter S. The fuctio vlues ƒsc k d re replced cotiuous selectio of fuctio vlues ƒ(). The suitervl widths k ecome the differetil d. It is s if we re summig ll products of the form ƒsd # d s goes from to. While this ottio cptures the process of costructig itegrl, it is Riem s defiitio tht gives precise meig to the defiite itegrl. Nottio d Eistece of the Defiite Itegrl The smol for the umer I i the defiitio of the defiite itegrl is ƒsd d which is red s the itegrl from to of ƒ of dee or sometimes s the itegrl from to of ƒ of with respect to. The compoet prts i the itegrl smol lso hve mes: Upper limit of itegrtio Itegrl sig ower limit of itegrtio Itegrl of f from to The fuctio is the itegrd. f() d is the vrile of itegrtio. Whe ou fid the vlue of the itegrl, ou hve evluted the itegrl. Copright 5 Perso Eductio, Ic., pulishig s Perso Addiso-Wesle

25 5.3 The Defiite Itegrl 35 Whe the defiitio is stisfied, we s the Riem sums of ƒ o [, ] coverge to the defiite itegrl I = ƒsd d d tht ƒ is itegrle over [, ]. We hve m choices for prtitio P with orm goig to zero, d m choices of poits c k for ech prtitio. The defiite itegrl eists whe we lws get the sme limit I, o mtter wht choices re mde. Whe the limit eists we write it s the defiite itegrl lim ƒƒpƒƒ: ƒsc k d k = I = ƒsd d. Whe ech prtitio hs equl suitervls, ech of width = s - d>, we will lso write lim : q ƒsc k d = I = ƒsd d. The limit is lws tke s the orm of the prtitios pproches zero d the umer of suitervls goes to ifiit. The vlue of the defiite itegrl of fuctio over prticulr itervl depeds o the fuctio, ot o the letter we choose to represet its idepedet vrile. If we decide to use t or u isted of, we simpl write the itegrl s ƒstd dt or ƒsud du isted of ƒsd d. No mtter how we write the itegrl, it is still the sme umer, defied s limit of Riem sums. Sice it does ot mtter wht letter we use, the vrile of itegrtio is clled dumm vrile. Sice there re so m choices to e mde i tkig limit of Riem sums, it might seem difficult to show tht such limit eists. It turs out, however, tht o mtter wht choices re mde, the Riem sums ssocited with cotiuous fuctio coverge to the sme limit. THEOREM The Eistece of Defiite Itegrls A cotiuous fuctio is itegrle. Tht is, if fuctio ƒ is cotiuous o itervl [, ], the its defiite itegrl over [, ] eists. B the Etreme Vlue Theorem (Theorem, Sectio.), whe ƒ is cotiuous we c choose c k so tht ƒsc k d gives the mimum vlue of ƒ o [ k -, k ], givig upper sum. We c choose c k to give the miimum vlue of ƒ o [ k -, k ], givig lower sum. We c pick c k to e the midpoit of [ k -, k ], the rightmost poit k, or rdom poit. We c tke the prtitios of equl or vrig widths. I ech cse we get the sme limit for g ƒsc k d k s 7P7 :. The ide ehid Theorem is tht Riem sum ssocited with prtitio is o more th the upper sum of tht prtitio d o less th the lower sum. The upper d lower sums coverge to the sme vlue whe 7P7 :. All other Riem sums lie etwee the upper d lower sums d hve the sme limit. A proof of Theorem ivolves creful lsis of fuctios, prtitios, d limits log this lie of thikig d is left to more dvced tet. A idictio of this proof is give i Eercises 8 d 8. Copright 5 Perso Eductio, Ic., pulishig s Perso Addiso-Wesle

26 36 Chpter 5: Itegrtio Theorem ss othig out how to clculte defiite itegrls. A method of clcultio will e developed i Sectio 5., through coectio to the process of tkig tiderivtives. Itegrle d Noitegrle Fuctios Theorem tells us tht fuctios cotiuous over the itervl [, ] re itegrle there. Fuctios tht re ot cotiuous m or m ot e itegrle. Discotiuous fuctios tht re itegrle iclude those tht re icresig o [, ] (Eercise 77), d the piecewise-cotiuous fuctios defied i the Additiol Eercises t the ed of this chpter. (The ltter re cotiuous ecept t fiite umer of poits i [, ].) For itegrilit to fil, fuctio eeds to e sufficietl discotiuous so tht the regio etwee its grph d the -is cot e pproimted well icresigl thi rectgles. Here is emple of fuctio tht is ot itegrle. EXAMPE A Noitegrle Fuctio o [, ] The fuctio hs o Riem itegrl over [, ]. Uderlig this is the fct tht etwee two umers there is oth rtiol umer d irrtiol umer. Thus the fuctio jumps up d dow too errticll over [, ] to llow the regio eeth its grph d ove the -is to e pproimted rectgles, o mtter how thi the re. We show, i fct, tht upper sum pproimtios d lower sum pproimtios coverge to differet limitig vlues. If we pick prtitio P of [, ] d choose c k to e the mimum vlue for ƒ o [ k -, k ] the the correspodig Riem sum is sice ech suitervl [ k -, k ] cotis rtiol umer where ƒsc k d =. Note tht the legths of the itervls i the prtitio sum to, g. So ech such Riem sum equls, d limit of Riem sums usig these choices equls. O the other hd, if we pick c k to e the miimum vlue for ƒ o [ k -, k ], the the Riem sum is sice ech suitervl [ k -, k ] cotis irrtiol umer c k where ƒsc k d =. The limit of Riem sums usig these choices equls zero. Sice the limit depeds o the choices of c k, the fuctio ƒ is ot itegrle. Properties of Defiite Itegrls I defiig ƒsd d s limit of sums g ƒsc k d k, we moved from left to right cross the itervl [, ]. Wht would hppe if we isted move right to left, strtig with = d edig t =. Ech k i the Riem sum would chge its sig, with k - k - ow egtive isted of positive. With the sme choices of c k i ech suitervl, the sig of Riem sum would chge, s would the sig of the limit, the itegrl ƒsd = e, if is rtiol, if is irrtiol U = = ƒsc k d k = ƒsc k d k = sd, sd k =, Copright 5 Perso Eductio, Ic., pulishig s Perso Addiso-Wesle

27 5.3 The Defiite Itegrl 37 ƒsd d. Sice we hve ot previousl give meig to itegrtig ckwrd, we re led to defie ƒsd d = - ƒsd d. Aother etesio of the itegrl is to itervl of zero width, whe =. Sice ƒsc k d k is zero whe the itervl width k =, we defie ƒsd d =. Theorem sttes seve properties of itegrls, give s rules tht the stisf, icludig the two ove. These rules ecome ver useful i the process of computig itegrls. We will refer to them repetedl to simplif our clcultios. Rules through 7 hve geometric iterprettios, show i Figure 5.. The grphs i these figures re of positive fuctios, ut the rules ppl to geerl itegrle fuctios. THEOREM Whe ƒ d g re itegrle, the defiite itegrl stisfies Rules to 7 i Tle 5.3. TABE 5.3 Rules stisfied defiite itegrls. Order of Itegrtio: ƒsd d = - ƒsd d A Defiitio. Zero Width Itervl: ƒsd d = Also Defiitio 3. Costt Multiple: kƒsd d = k ƒsd d A Numer k -ƒsd d = - ƒsd d k = -. Sum d Differece: sƒsd ; gsdd d = ƒsd d ; gsd d 5. Additivit: c c ƒsd d + ƒsd d = ƒsd d 6. M-Mi Iequlit: If ƒ hs mimum vlue m ƒ d miimum vlue mi ƒ o [, ], the 7. Domitio: mi ƒ # s - d ƒsd d m ƒ # s - d. ƒsd Ú gsd o [, ] Q ƒsd d Ú gsd d ƒsd Ú o [, ] Q ƒsd d Ú (Specil Cse) Copright 5 Perso Eductio, Ic., pulishig s Perso Addiso-Wesle

28 38 Chpter 5: Itegrtio f () f () f () f () g() g() f () () Zero Width Itervl: ƒsd d =. (The re over poit is.) () Costt Multiple: kƒsd d = k ƒsd d. (Show for k =. ) (c) Sum: sƒsd + gsdd d = ƒsd d + gsd d (Ares dd) f() f() d f () d c c m f mi f f () f () g() (d) Additivit for defiite itegrls: c c ƒsd d + ƒsd d = ƒsd d FIGURE 5. (e) M-Mi Iequlit: mi ƒ # s - d ƒsd d m ƒ # s - d (f ) Domitio: ƒsd Ú gsd o [, ] Q ƒsd d Ú gsd d While Rules d re defiitios, Rules 3 to 7 of Tle 5.3 must e proved. The proofs re sed o the defiitio of the defiite itegrl s limit of Riem sums. The followig is proof of oe of these rules. Similr proofs c e give to verif the other properties i Tle 5.3. Proof of Rule 6 Rule 6 ss tht the itegrl of ƒ over [, ] is ever smller th the miimum vlue of ƒ times the legth of the itervl d ever lrger th the mimum vlue of ƒ times the legth of the itervl. The reso is tht for ever prtitio of [, ] d for ever choice of the poits c k, mi ƒ # s - d = mi ƒ # k = mi ƒ # k ƒsc k d k m ƒ # k = m ƒ # k = m ƒ # s - d. k = - Costt Multiple Rule mi ƒ ƒsc k d ƒsc k d m f Costt Multiple Rule Copright 5 Perso Eductio, Ic., pulishig s Perso Addiso-Wesle

29 5.3 The Defiite Itegrl 39 I short, ll Riem sums for ƒ o [, ] stisf the iequlit Hece their limit, the itegrl, does too. EXAMPE Usig the Rules for Defiite Itegrls Suppose tht ƒsd d = 5, ƒsd d = -, hsd d = The. ƒsd d = - ƒsd d = -s -d = Rule. [ƒsd + 3hsd] d = ƒsd d + 3 hsd d Rules 3 d = s5d + 3s7d = 3 3. ƒsd d = ƒsd d + ƒsd d = 5 + s -d = Rule 5 EXAMPE 3 Fidig Bouds for Itegrl Show tht the vlue of + cos d is less th 3>. Solutio The M-Mi Iequlit for defiite itegrls (Rule 6) ss tht mi ƒ # s - d is lower oud for the vlue of d tht m ƒ # ƒsd d s - d is upper oud. The mimum vlue of + cos o [, ] is + =, so Sice + cos d is ouded from ove (which is. Á ), the itegrl is less th 3>. mi ƒ # s - d ƒsc k d k m ƒ # s - d. + cos d # s - d =. Are Uder the Grph of Noegtive Fuctio We ow mke precise the otio of the re of regio with curved oudr, cpturig the ide of pproimtig regio icresigl m rectgles. The re uder the grph of oegtive cotiuous fuctio is defied to e defiite itegrl. DEFINITION Are Uder Curve s Defiite Itegrl If = ƒsd is oegtive d itegrle over closed itervl [, ], the the re uder the curve = ƒsd over [, ] is the itegrl of ƒ from to, A = ƒsd d. Copright 5 Perso Eductio, Ic., pulishig s Perso Addiso-Wesle

30 35 Chpter 5: Itegrtio For the first time we hve rigorous defiitio for the re of regio whose oudr is the grph of cotiuous fuctio. We ow ppl this to simple emple, the re uder stright lie, where we c verif tht our ew defiitio grees with our previous otio of re. FIGURE 5. The regio i Emple is trigle. EXAMPE Are Uder the ie = Compute d d fid the re A uder = over the itervl [, ], 7. Solutio The regio of iterest is trigle (Figure 5.). We compute the re i two ws. () To compute the defiite itegrl s the limit of Riem sums, we clculte lim ƒƒpƒƒ: g ƒsc k d k for prtitios whose orms go to zero. Theorem tells us tht it does ot mtter how we choose the prtitios or the poits c k s log s the orms pproch zero. All choices give the ect sme limit. So we cosider the prtitio P tht sudivides the itervl [, ] ito suitervls of equl width = s - d> = >, d we choose to e the right edpoit i ech suitervl. The prtitio is c k P = e,,, 3, Á, f d c k = k. So ƒsc k d = Costt Multiple Rule Sum of First Itegers As : q d 7P7 :, this lst epressio o the right hs the limit >. Therefore, = = = = s + d d =. # s + d () Sice the re equls the defiite itegrl for oegtive fuctio, we c quickl derive the defiite itegrl usig the formul for the re of trigle hvig se legth d height =. The re is A = s>d # = >. Agi we hve tht d = >. k # k k ƒsc k d = c k Emple c e geerlized to itegrte ƒsd = [, ], 6 6. d = d + d = - d + d =- +. over closed itervl Rule 5 Rule Emple Copright 5 Perso Eductio, Ic., pulishig s Perso Addiso-Wesle

31 5.3 The Defiite Itegrl 35 I coclusio, we hve the followig rule for itegrtig f() = : d = -, 6 () This computtio gives the re of trpezoid (Figure 5.3). Equtio () remis vlid whe d re egtive. Whe 6 6, the defiite itegrl vlue s - d> is egtive umer, the egtive of the re of trpezoid droppig dow to the lie = elow the -is. Whe 6 d 7, Equtio () is still vlid d the defiite itegrl gives the differece etwee two res, the re uder the grph d ove [, ] mius the re elow [, ] d over the grph. The followig results c lso e estlished usig Riem sum clcultio similr to tht i Emple (Eercises 75 d 76). FIGURE 5.3 The re of this trpezoidl regio is A = s - d>. c d = cs - d, c costt d = , 6 () (3) f() (c k, f(c k )) c k FIGURE 5. A smple of vlues of fuctio o itervl [, ]. Averge Vlue of Cotiuous Fuctio Revisited I Sectio 5. we itroduced iformll the verge vlue of oegtive cotiuous fuctio ƒ over itervl [, ], ledig us to defie this verge s the re uder the grph of = ƒsd divided -. I itegrl ottio we write this s We c use this formul to give precise defiitio of the verge vlue of cotiuous (or itegrle) fuctio, whether positive, egtive or oth. Altertel, we c use the followig resoig. We strt with the ide from rithmetic tht the verge of umers is their sum divided. A cotiuous fuctio ƒ o [, ] m hve ifiitel m vlues, ut we c still smple them i orderl w. We divide [, ] ito suitervls of equl width = s - d> d evlute ƒ t poit i ech (Figure 5.). The verge of the smpled vlues is c k Averge = ƒsc d + ƒsc d + Á + ƒsc d = ƒsc k d = - ƒsc k d = ƒsd d. - - ƒsc k d = -, so = - Copright 5 Perso Eductio, Ic., pulishig s Perso Addiso-Wesle

32 35 Chpter 5: Itegrtio The verge is otied dividig Riem sum for ƒ o [, ] s - d. As we icrese the size of the smple d let the orm of the prtitio pproch zero, the verge pproches (>( - )) ƒsd d. Both poits of view led us to the followig defiitio. DEFINITION The Averge or Me Vlue of Fuctio If ƒ is itegrle o [, ], the its verge vlue o [, ], lso clled its me vlue, is vsƒd = ƒsd d. - f() FIGURE 5.5 The verge vlue of ƒsd = - o [-, ] is p> (Emple 5). EXAMPE 5 Fidig Averge Vlue Fid the verge vlue of ƒsd = - o [-, ]. Solutio We recogize ƒsd = - s fuctio whose grph is the upper semicircle of rdius cetered t the origi (Figure 5.5). The re etwee the semicircle d the -is from - to c e computed usig the geometr formul Becuse ƒ is oegtive, the re is lso the vlue of the itegrl of ƒ from - to, Therefore, the verge vlue of ƒ is vsƒd = Are = # pr = # psd = p. - - d = p. - d = - s -d spd = p. - Copright 5 Perso Eductio, Ic., pulishig s Perso Addiso-Wesle