AW/Thoms_ch-9 8// 9:7 AM Pge 6 6 Chter : Itegrtio (, ) (, ), 6,,., 7 6. R R....7 () () FIGURE. () We get uer estimte of the re of R usig two rectgles

Transcription

1 AW/Thoms_ch-9 8// 9:7 AM Pge Chter INTEGRATION OVERVIEW Oe of the gret chievemets of clssicl geometr ws to oti formuls for the res d volumes of trigles, sheres, d coes. I this chter we stud method to clculte the res d volumes of these d other more geerl shes. The method we develo, clled itegrtio, is tool for clcultig much more th res d volumes. The itegrl hs m lictios i sttistics, ecoomics, the scieces, d egieerig. It llows us to clculte qutities rgig from roilities d verges to eerg cosumtio d the forces gist dm s floodgtes. The ide ehid itegrtio is tht we c effectivel comute m qutities rekig them ito smll ieces, d the summig the cotriutios from ech smll rt. We develo the theor of the itegrl i the settig of re, where it most clerl revels its ture. We egi with emles ivolvig fiite sums. These led turll to the questio of wht hes 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.. The ture of their coectio, cotied i the Fudmetl Theorem of Clculus, is oe of the most imortt ides i clculus. Estimtig with Fiite Sums. This sectio shows how re, verge vlues, d the distce trveled oject over time c ll e roimted fiite sums. Fiite sums re the sis for defiig the itegrl i Sectio.. Are The re of regio with curved oudr c e roimted summig the res of collectio of rectgles. Usig more rectgles c icrese the ccurc of the roimtio.. R EXAMPE. FIGURE. The re of the regio R cot e foud simle geometr formul (Emle ). Aroimtig Are Wht is the re of the shded regio R tht lies ove the -is, elow the grh of -, d etwee the verticl lies d? (See Figure..) A rchitect might wt to kow this re to clculte the weight of custom widow with she descried R. Ufortutel, there is o simle geometric formul for clcultig the res of shes hvig curved oudries like the regio R. Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle

2 AW/Thoms_ch-9 8// 9:7 AM Pge 6 6 Chter : Itegrtio (, ) (, ), 6,,., 7 6. R R....7 () () FIGURE. () We get uer estimte of the re of R usig two rectgles cotiig R. () Four rectgles give etter uer 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 roimte it i simle w. Figure. shows two rectgles tht together coti the regio R. Ech rectgle hs width > d the hve heights d >, movig from left to right. The height of ech rectgle is the mimum vlue of the fuctio ƒ, otied evlutig ƒ t the left edoit of the suitervl of [, ] formig the se of the rectgle. The totl re of the two rectgles roimtes the re A of the regio R, A # + # This estimte is lrger th the true re A, sice the two rectgles coti R. We s tht.87 is uer sum ecuse it is otied tkig the height of ech rectgle s the mimum (uermost) vlue of ƒ() for oit i the se itervl of the rectgle. I Figure., we imrove our estimte usig four thier rectgles, ech of width >, which tke together coti the regio R. These four rectgles give the roimtio A # # # #.78, which is still greter th A sice the four rectgles coti R. Suose isted we use four rectgles cotied iside the regio R to estimte the re, s i Figure.. Ech rectgle hs width > s efore, ut the rectgles re shorter d lie etirel eeth the grh 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 edoit 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 oit i ech se suitervl, gives lower sum roimtio to the re, A 6 # + # # 7 + #.. 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 uer sums:. 6 A Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle

3 AW/Thoms_ch-9 8// 9:7 AM Pge 7 7. Estimtig with Fiite Sums, 6, 6 8 6, 8 6,, 9 8 6, , () () FIGURE. () Rectgles cotied i R give estimte for the re tht udershoots the true vlue. () The midoit rule uses rectgles whose height is the vlue of ƒsd t the midoits of their ses. () B cosiderig oth lower d uer sum roimtios we get ot ol estimtes for the re, ut lso oud o the size of the ossile error i these estimtes sice the true vlue of the re lies somewhere etwee them. Here the error cot e greter th the differece Yet other estimte c e otied usig rectgles whose heights re the vlues of ƒ t the midoits of their ses (Figure.). This method of estimtio is clled the midoit rule for roimtig the re. The midoit rule gives estimte tht is etwee lower sum d uer sum, ut it is ot cler whether it overestimtes or uderestimtes the true re. With four rectgles of width > s efore, the midoit rule estimtes the re of R to e A # + 6 # # + 6 # 7 6 # I ech of our comuted 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 oit i ech suitervl: c i the first suitervl, c i the secod suitervl, d so o. The fiite sums the ll tke the form 6 6 ƒsc d + ƒsc d + ƒsc d + Á + ƒsc d. () FIGURE. () A lower sum usig 6 rectgles of equl width >6. () A uer sum usig 6 rectgles. B tkig more d more rectgles, with ech rectgle thier th efore, it ers tht these fiite sums give etter d etter roimtios to the true re of the regio R. Figure. shows lower sum roimtio for the re of R usig 6 rectgles of equl width. The sum of their res is.6766, which ers close to the true re, ut is still smller sice the rectgles lie iside R. Figure. shows uer sum roimtio usig 6 rectgles of equl width. The sum of their res is.69766, which is somewht lrger th the true re ecuse the rectgles tke together coti R. The midoit rule for 6 rectgles gives totl re roimtio of , ut it is ot immeditel cler whether this estimte is lrger or smller th the true re. Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle

4 AW/Thoms_ch-9 8// 9:7 AM Pge 8 8 Chter : Itegrtio TABE. Fiite roimtios for the re of R Numer of suitervls ower sum Midoit rule Uer sum Tle. shows the vlues of uer d lower sum roimtios to the re of R usig u to rectgles. I Sectio. 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 develoed there, we will e le to show tht the re of R is ectl >. Distce Trveled Suose 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) we c fid the cr s ositio fuctio s(t) settig sstd Fstd + C. The distce trveled c the e foud clcultig the chge i ositio, ssd - ssd (see Eercise 9, Sectio.8). If the velocit fuctio is determied recordig seedometer 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 roimte 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 roimte the distce trveled o ech time suitervl with the usul distce formul distce velocit * time d dd the results cross [, ]. Suose the sudivided itervl looks like t t t t t t 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. Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle

5 AW/Thoms_ch-9 8// 9:7 AM Pge 9. Estimtig with Fiite Sums EXAMPE 9 Estimtig the Height of Projectile The velocit fuctio of rojectile fired stright ito the ir is ƒstd 6-9.8t m>sec. Use the summtio techique just descried to estimte how fr the rojectile rises durig the first sec. How close do the sums come to the ect figure of.9 m? We elore the results for differet umers of itervls d differet choices of evlutio oits. Notice tht ƒ(t) is decresig, so choosig left edoits gives uer sum estimte; choosig right edoits gives lower sum estimte. Solutio () Three suitervls of legth, with ƒ evluted t left edoits givig uer sum: t t t t t With ƒ evluted t t,, d, we hve D ƒst d t + ƒst d t + ƒst d t [6-9.8sd]sd + [6-9.8sd]sd + [6-9.8sd]sd.6. () Three suitervls of legth, with ƒ evluted t right edoits givig lower sum: t t t t t With ƒ evluted t t,, d, we hve D ƒst d t + ƒst d t + ƒst d t [6-9.8sd]sd + [6-9.8sd]sd + [6-9.8sd]sd.. (c) With si suitervls of legth >, we get t t t t t t 6 t t t t t t t 6 t t t A uer sum usig left edoits: D.; lower sum usig right edoits: D 8.. These si-itervl estimtes re somewht closer th the three-itervl estimtes. The results imrove s the suitervls get shorter. As we c see i Tle., the left-edoit uer sums roch the true vlue.9 from ove, wheres the right-edoit lower sums roch it from elow. The true Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle

6 AW/Thoms_ch-9 8// 9:7 AM Pge Chter : Itegrtio TABE. Trvel-distce estimtes Numer of suitervls egth of ech suitervl Uer sum ower sum > > >8 >6 > > vlue lies etwee these uer d lower sums. The mgitude of the error i the closest etries is., smll ercetge of the true vlue. Error mgitude ƒ true vlue - clculted vlue ƒ ƒ ƒ.. Error ercetge..%..9 It would e resole to coclude from the tle s lst etries tht the rojectile rose out 6 m durig its first sec of flight. Dislcemet Versus Distce Trveled If od with ositio 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 Emle. If the od chges directio oe or more times durig the tri, the we eed to use the od s seed 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 Emle, ol gives estimte to the od s dislcemet, ssd - ssd, the differece etwee its iitil d fil ositios. To see wh, rtitio the time itervl [, ] ito smll eough equl suitervls t so tht the od s velocit does ot chge ver much from time tk - to tk. The stk d gives good roimtio of the velocit throughout the itervl. Accordigl, the chge i the od s ositio coordite durig the time itervl is out stk d t. The chge is ositive if stk d is ositive d egtive if stk d is egtive. I either cse, the distce trveled durig the suitervl is out ƒ stk d ƒ t. The totl distce trveled is roimtel the sum ƒ st d ƒ t + ƒ st d ƒ t + Á + ƒ st d ƒ t. Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle

7 AW/Thoms_ch-9 8// 9:7 AM Pge. Estimtig with Fiite Sums g() c c () c () FIGURE. () 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 grh divided -. 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 emle, the temerture t certi loctio i tow is cotiuous fuctio tht goes u d dow ech d. Wht does it me to s tht the verge temerture i the tow over the course of d is 7 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 ositive, its grh over [, ] gives rectgle of height c. The verge vlue of the fuctio c the e iterreted geometricll s the re of this rectgle divided its width - (Figure.). Wht if we wt to fid the verge vlue of ocostt fuctio, such s the fuctio g i Figure.? We c thik of this grh s sshot 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 oit 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 grh of g divided -. We re led to defie the verge vlue of oegtive fuctio o itervl [, ] to e the re uder its grh divided -. For this defiitio to e vlid, we eed recise uderstdig of wht is met the re uder grh. This will e otied i Sectio., ut for ow we look t two simle emles. 6 f() EXAMPE Wht is the verge vlue of the fuctio ƒsd o the itervl [, ]? Solutio The verge equls the re uder the grh divided the width of the itervl. I this cse we do ot eed fiite roimtio to estimte the re of the regio uder the grh: trigle of height 6 d se hs re 6 (Figure.6). The width of the itervl is - -. The verge vlue of the fuctio is 6>. The Averge Vlue of ier Fuctio EXAMPE The Averge Vlue of si Estimte the verge vlue of the fuctio ƒsd si o the itervl [, ]. FIGURE.6 The verge vlue of ƒsd over [, ] is (Emle ). ookig t the grh of si etwee d i Figure.7, we c see tht its verge height is somewhere etwee d. To fid the verge we eed to Solutio Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle

8 AW/Thoms_ch-9 8// 9:7 AM Pge Chter : Itegrtio f () si f () si () () FIGURE.7 Aroimtig the re uder ƒsd si etwee d to comute the verge vlue of si over [, ], usig () four rectgles; () eight rectgles (Emle ). clculte the re A uder the grh d the divide this re the legth of the itervl, -. We do ot hve simle w to determie the re, so we roimte it with fiite sums. To get uer sum estimte, we dd the res of four rectgles of equl width > tht together coti the regio eeth the grh of si d ove the -is o [, ]. We choose the heights of the rectgles to e the lrgest vlue of si o ech suitervl. Over rticulr suitervl, this lrgest vlue m occur t the left edoit, the right edoit, or somewhere etwee them. We evlute si t this oit to get the height of the rectgle for uer sum. The sum of the rectgle res the estimtes the totl re (Figure.7): A si + si # + si # s.d #.69. # # + si # To estimte the verge vlue of si we divide the estimted re d oti the roimtio.69>.86. If we use eight rectgles of equl width >8 ll lig ove the grh of si (Figure.7), we get the re estimte A si 7 # + si + si + si + si + si + si + si s6.d #.6. s d # 8 8 Dividig this result the legth of the itervl gives more ccurte estimte of.7 for the verge. Sice we used uer sum to roimte the re, this estimte is still greter th the ctul verge vlue of si over [, ]. 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., we will show tht the true verge vlue is >.6. As efore, we could just s well hve used rectgles lig uder the grh of si d clculted lower sum roimtio, or we could hve used the midoit rule. I Sectio., we will see tht it does t mtter whether our roimtig rectgles re chose to give uer sums, lower sums, or sum i etwee. I ech cse, the roimtios re close to the true re if ll the rectgles re sufficietl thi. Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle

9 AW/Thoms_ch-9 8// 9:7 AM Pge. Estimtig with Fiite Sums Summr The re uder the grh of ositive fuctio, the distce trveled movig oject tht does t chge directio, d the verge vlue of oegtive fuctio over itervl c ll e roimted fiite sums. First we sudivide the itervl ito suitervls, tretig the rorite fuctio ƒ s if it were costt over ech rticulr suitervl. The we multil the width of ech suitervl the vlue of ƒ t some oit withi it, d dd these roducts together. If the itervl [, ] is sudivided ito suitervls of equl widths s - d>, d if ƒsck d is the vlue of ƒ t the chose oit ck i the kth suitervl, this rocess gives fiite sum of the form ƒsc d + ƒsc d + ƒsc d + Á + ƒsc d. The choices for the ck 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 roimtios give uer sums d lower sums. The fiite sum roimtios we looked t imroved s we took more suitervls of thier width. Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle

10 AW/Thoms_ch-9 8// 9:7 AM Pge. Estimtig with Fiite Sums EXERCISES. Are I Eercises use fiite roimtios to estimte the re uder the grh of the fuctio usig. lower sum with two rectgles of equl width.. lower sum with four rectgles of equl width. c. uer sum with two rectgles of equl width. d. uer sum with four rectgles of equl width.. ƒsd etwee d. Time (sec) Velocit (i. / sec) Time (sec) Velocit (i. / sec) ƒsd etwee d.. ƒsd > etwee d.. ƒsd - etwee - d. Usig rectgles whose height is give the vlue of the fuctio t the midoit of the rectgle s se (the midoit rule) estimte the re uder the grhs of the followig fuctios, usig first two d the four rectgles.. ƒsd etwee d. 6. ƒsd etwee d.. Distce trveled ustrem You re sittig o the k of tidl river wtchig the icomig tide crr ottle ustrem. You record the velocit of the flow ever miutes for hour, with the results show i the ccomig tle. Aout how fr ustrem did the ottle trvel durig tht hour? Fid estimte usig suitervls of legth with. left-edoit vlues.. right-edoit vlues. 7. ƒsd > etwee d. 8. ƒsd - etwee - d. Distce 9. Distce trveled The ccomig 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-edoit vlues.. right-edoit vlues. Time (mi) Velocit (m / sec) Time (mi) Velocit (m / sec) Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle

11 AW/Thoms_ch-9 8// 9:7 AM Pge Chter : Itegrtio. egth of rod You d comio re out to drive twist stretch of dirt rod i cr whose seedometer works ut whose odometer (milege couter) is roke. To fid out how log this rticulr stretch of rod is, ou record the cr s velocit t -sec itervls, with the results show i the ccomig tle. Estimte the legth of the rod usig. left-edoit vlues.. right-edoit vlues. Time (sec) Velocit (coverted to ft / sec) ( mi / h ft / sec) 6 Time (sec) Velocit (coverted to ft / sec) ( mi / h ft / sec) Use rectgles to estimte how fr the cr trveled durig the 6 sec it took to rech mi> h.. Roughl how m secods did it tke the cr to rech the hlfw oit? Aout how fst ws the cr goig the? Velocit d Distce. Free fll with ir resistce A oject is droed stright dow from helicoter. 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 dro for sec, s show: t Fid uer estimte for the seed whe t.. Fid lower estimte for the seed whe t. c. Fid uer estimte for the distce flle whe t.. Distce trveled rojectile A oject is shot stright uwrd from se level with iitil velocit of ft> sec.. Distce from velocit dt The ccomig tle gives dt for the velocit of vitge sorts cr ccelertig from to mi> h i 6 sec ( thousdths of hour). Time (h) Velocit (mi / h) Time (h) Velocit (mi / h) Assumig tht grvit is the ol force ctig o the oject, give uer estimte for its velocit fter sec hve elsed. Use g ft>sec for the grvittiol ccelertio.. Fid lower estimte for the height ttied fter sec. Averge Vlue of Fuctio I Eercises 8, use fiite sum to estimte the verge vlue of ƒ o the give itervl rtitioig the itervl ito four suitervls of equl legth d evlutig ƒ t the suitervl midoits.. ƒsd 6. ƒsd > o [, 9] o [, ] 7. ƒstd s>d + si t o [, ] si t. mi/hr ƒstd - cos t t o [, ] 8 cos t hours Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle t

12 AW/Thoms_ch-9 8// 9:7 AM Pge. Estimtig with Fiite Sums Pollutio Cotrol 9. Wter ollutio 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) ekge (gl> h) Time (h) ekge (gl> h) c. The tker cotiues to lek 7 gl> h fter the first 8 hours. If the tker origill cotied, gl of oil, roimtel how m more hours will else i the worst cse efore ll the oil hs silled? I the est cse?. Air ollutio A ower lt geertes electricit urig oil. Pollutts roduced s result of the urig rocess re removed scruers i the smokestcks. Over time, the scruers ecome less efficiet d evetull the must e relced whe the mout of ollutio relesed eceeds govermet stdrds. Mesuremets re tke t the ed of ech moth determiig the rte t which ollutts re relesed ito the tmoshere, recorded s follows. Moth J Fe Mr Ar M Ju Pollutt Relese rte (tos> d) Moth Jul Aug Se Oct Nov Dec I the est cse, roimtel whe will totl of tos of ollutts hve ee relesed ito the tmoshere? Are of Circle. (squre). Reet rt () for the qutit of oil tht hs esced fter 8 hours..6. Assumig -d moth d tht ew scruers llow ol. to> d relesed, give uer estimte of the totl toge of ollutts relesed the ed of Jue. Wht is lower estimte?. Iscrie regulr -sided olgo iside circle of rdius d comute the re of the olgo for the followig vlues of :. Give uer d lower estimte of the totl qutit of oil tht hs esced fter hours. Pollutt Relese rte (tos> d).9. 8 (octgo) c. 6 d. Comre the res i rts (), (), d (c) with the re of the circle.. (Cotiutio of Eercise ). Iscrie regulr -sided olgo iside circle of rdius d comute the re of oe of the cogruet trigles formed drwig rdii to the vertices of the olgo.. Comute the limit of the re of the iscried olgo s : q. c. Reet the comuttios i rts () d () for circle of rdius r. COMPUTER EXPORATIONS I Eercises 6, use CAS to erform the followig stes.. Plot the fuctios over the give itervl.. Sudivide the itervl ito,, d suitervls of equl legth d evlute the fuctio t the midoit of ech suitervl. c. Comute the verge vlue of the fuctio vlues geerted i rt (). d. Solve the equtio ƒsd sverge vlued for usig the verge vlue clculted i rt (c) for the rtitioig.. ƒsd si o [, ]. ƒsd si 6. ƒsd si o o. ƒsd si c, d c, d Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle o [, ]

13 AW/Thoms_ch-9 8// 9:7 AM Pge. Sigm Nottio d imits of Fiite Sums. Sigm Nottio d imits of Fiite Sums I estimtig with fiite sums i Sectio., we ofte ecoutered sums with m terms (u to i Tle., 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 roerties, we look t wht hes to fiite sum roimtio s the umer of terms roches ifiit. CoGh

14 AW/Thoms_ch-9 8// 9:7 AM Pge 6 6 Chter : Itegrtio Fiite Sums d Sigm Nottio Sigm ottio eles us to write sum with m terms i the comct form Á k k The Greek letter (citl sigm, corresodig 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 ide k eds t k. The summtio smol (Greek letter sigm) k k is formul for the kth term. k The ide k strts t k. Thus we c write k, k d ƒsd + ƒsd + ƒsd + Á + ƒsd ƒsid. i The sigm ottio used o the right side of these equtios is much more comct th the summtio eressios o the left side. EXAMPE Usig Sigm Nottio The sum i sigm ottio The sum writte out, oe term for ech vlue of k The vlue of the sum k k s - d k s - dsd + s - dsd + s - dsd k k + k k k k k - k The lower limit of summtio does ot hve to e ; it c e iteger. Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle

15 AW/Thoms_ch-9 8// 9:7 AM Pge 7. Sigm Nottio d imits of Fiite Sums EXAMPE 7 Usig Differet Ide Strtig Vlues Eress 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 simlest to strt with k or k sk + d Strtig with k : k sk - d Strtig with k : k sk - d Strtig with k : k sk + 7d Strtig with k - : k - Whe we hve sum such s sk + k d k we c rerrge its terms, sk + k d s + d + s + d + s + d k s + + d + s + + d Regrou terms. k + k k k This illustrtes geerl rule for fiite sums: sk + k d k + k k k k Four such rules re give elow. A roof tht the re vlid c e otied usig mthemticl iductio (see Aedi ). Alger Rules for Fiite Sums. Sum Rule: k. Differece Rule: Costt Multile Rule: Costt Vlue Rule: k k # c c k # ck c k k. k (k - k) k - k k. (k + k) k + k (A umer c) k (c is costt vlue.) k Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle

16 AW/Thoms_ch-9 8// 9:7 AM Pge 8 8 Chter : Itegrtio EXAMPE Usig the Fiite Sum Alger Rules Differece Rule d Costt Multile Rule () sk - k d k - k k k k Costt Multile Rule () s - k d s - d # k - # k - k k k k k (c) sk + d k + k k Sum Rule k s + + d + s # d Costt Vlue Rule (d) # HISTORICA BIOGRAPHY Crl Friedrich Guss (777 8) Costt Vlue Rule (> is costt) k Over the ers eole 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 k k Solutio: s + d. The formul tells us tht the sum of the first itegers is sdsd. Additio verifies this redictio: To rove the formul i geerl, we write out the terms i the sum twice, oce forwrd d oce ckwrd. + + s - d + + s - d + + Á 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 roved usig mthemticl iductio (see Aedi ). We stte them here. The first squres: k k The first cues: s + ds + d 6 k k s + d Coright Perso EduÍç@ QQ ãû, Rý ÚdH-wØüß m dd GÌ^Ã#lÛ;¼øÅJ áhdì

17 AW/Thoms_ch-9 8// 9:7 AM Pge 9. Sigm Nottio d imits of Fiite Sums 9 imits of Fiite Sums The fiite sum roimtios we cosidered i Sectio. got more ccurte s the umer of terms icresed d the suitervl widths (legths) ecme thier. The et emle shows how to clculte limitig vlue s the widths of the suitervls go to zero d their umer grows to ifiit. EXAMPE The imit of Fiite Aroimtios to Are Fid the limitig vlue of lower sum roimtios to the re of the regio R elow the grh of - d ove the itervl [, ] o the -is usig equl width rectgles whose widths roch zero d whose umer roches ifiit. (See Figure..) We comute lower sum roimtio usig rectgles of equl width s - d>, d the we see wht hes s : q. We strt sudividig [, ] ito equl width suitervls Solutio - c, d, c, d, Á, c, d. Ech suitervl hs width >. The fuctio - is decresig o [, ], d its smllest vlue i suitervl occurs t the suitervl s right edoit. So lower sum is costructed with rectgles whose height over the suitervl [sk - d>, k>] is ƒsk>d - sk>d, givig the sum k ƒ + ƒ + Á + ƒ + Á + ƒ. We write this i sigm ottio d simlif, k k k k ƒ - k - k k - k k # - k k sds + ds + d Differece Rule Costt Vlue d Costt Multile Rules Sum of the First Squres Numertor eded We hve otied eressio for the lower sum tht holds for. Tkig the limit of this eressio s : q, we see tht the lower sums coverge s the umer of suitervls icreses d the suitervl widths roch zero: lim - : q The lower sum roimtios coverge to >. A similr clcultio shows tht the uer sum roimtios lso coverge to > (Eercise ). A fiite sum roimtio, i the sese of our summr t the ed of Sectio., lso coverges to the sme vlue Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle

18 AW/Thoms_ch-9 8// 9:7 AM Pge Chter : Itegrtio f () FIGURE.8 A ticl cotiuous fuctio ƒsd over closed itervl [, ]. >. This is ecuse it is ossile to show tht fiite sum roimtio is tred etwee the lower d uer sum roimtios. For this reso we re led to defie the re of the regio R s this limitig vlue. I Sectio. we stud the limits of such fiite roimtios i their more geerl settig. Riem Sums HISTORICA BIOGRAPHY Georg Friedrich Berhrd Riem (86 866) The theor of limits of fiite roimtios ws mde recise 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 ictured i Figure.8, ƒ m hve egtive s well s ositive 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 roimtios i Sectio.. To do so, we choose - oits,,, Á, - 6 etwee d d stisfig Á 6-6. To mke the ottio cosistet, we deote d, so tht Á 6-6. The set P,,, Á, -, 6 is clled rtitio of [, ]. The rtitio P divides [, ] ito closed suitervls [, ], [, ], Á, [ -, ]. The first of these suitervls is [, ], the secod is [, ], d the k th suitervl of P is [k -, k], for k iteger etwee d. Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle

19 AW/Thoms_ch-9 8// 9:7 AM Pge. Sigm Nottio d imits of Fiite Sums 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 oit. The oit chose i the kth suitervl [k -, k] is clled ck. The o ech suitervl we std verticl rectgle tht stretches from the -is to touch the curve t sck, ƒsck dd. These rectgles c e ove or elow the -is, deedig o whether ƒsck d is ositive or egtive, or o it if ƒsck d (Figure.9). O ech suitervl we form the roduct ƒsck d # k. This roduct is ositive, egtive or zero, deedig o the sig of ƒsck d. Whe ƒsck d 7, the roduct ƒsck d # k is the re of rectgle with height ƒsck d d width k. Whe ƒsck d 6, the roduct ƒsck d # k is egtive umer, the egtive of the re of rectgle of width k tht dros from the -is to the egtive umer ƒsck d. Fill we sum ll these roducts to get SP ƒsck d k. k f () (c, f(c )) (ck, f (ck )) kth rectgle c c k ck k c (c, f (c)) (c, f (c )) FIGURE.9 The rectgles roimte the regio etwee the grh of the fuctio ƒsd d the -is. Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle

20 AW/Thoms_ch-9 8// 9:7 AM Pge Chter : Itegrtio f () The sum SP is clled Riem sum for ƒ o the itervl [, ]. There re m such sums, deedig o the rtitio P we choose, d the choices of the oits ck i the suitervls. I Emle, where the suitervls ll hd equl widths >, we could mke them thier siml icresig their umer. Whe rtitio hs suitervls of vrig widths, we c esure the re ll thi cotrollig the width of widest (logest) suitervl. We defie the orm of rtitio P, writte 7P7, to e the lrgest of ll the suitervl widths. If 7P7 is smll umer, the ll of the suitervls i the rtitio P hve smll width. et s look t emle of these ides. EXAMPE 6 () The set P {,.,.6,,., } is rtitio of [, ]. There re five suitervls of P: [,.], [.,.6], [.6, ], [,.], d [., ]: f() Prtitioig Closed Itervl..6. The legths of the suitervls re.,.,.,., d.. The logest suitervl legth is., so the orm of the rtitio is 7P7.. I this emle, there re two suitervls of this legth. () FIGURE. The curve of Figure.9 with rectgles from fier rtitios of [, ]. Fier rtitios crete collectios of rectgles with thier ses tht roimte the regio etwee the grh of ƒ d the -is with icresig ccurc. A Riem sum ssocited with rtitio of closed itervl [, ] defies rectgles tht roimte the regio etwee the grh of cotiuous fuctio ƒ d the -is. Prtitios with orm rochig zero led to collectios of rectgles tht roimte this regio with icresig ccurc, s suggested Figure.. 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 rtitio P d the oits ck i its suitervls to costruct Riem sum, sigle limitig vlue is roched s the suitervl widths, cotrolled the orm of the rtitio, roch zero. Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle

21 AW/Thoms_ch-9 8// 9:7 AM Pge Chter : Itegrtio EXERCISES. Sigm Nottio Write the sums i Eercises 6 without sigm ottio. The evlute them. 6k. k k +. k. s - dk + si k k 6. s -dk cos k k k 7. Which of the followig eress i sigm ottio? 6 k. k k. s - dk k k c. s - dk + k + k - 9. Which formul is ot equivlet to the other two? s - dk - s - dk s - dk.. c. k k - k k + k - k +. si k. k - 6. s - dk - k k - k. cos k k 8. Which of the followig eress i sigm ottio?. Which formul is ot equivlet to the other two?. sk - d k. sk + d k - c. k + k - Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle - c. k k -

22 AW/Thoms_ch-9 8// 9:7 AM Pge. Sigm Nottio d imits of Fiite Sums Eress the sums i Eercises 6 i sigm ottio. The form of our swer will deed o our choice of the lower limit of summtio Vlues of Fiite Sums 7. Suose tht k - d k 6. Fid the vlues of k k k. k 6. k k c. sk + k d k d. sk - k d k e. sk - k d 8. Suose tht k d k. Fid the vlues of k k 7 6. ksk + d. ksk + d k k + k 7. k k k 7 7 k 8. k - k k Rectgles for Riem Sums I Eercises 9, grh 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 k ƒsck d k, give tht ck is the () left-hd edoit, () righthd edoit, (c) midoit of the kth suitervl. (Mke serte sketch for ech set of rectgles.) 9. ƒsd -, [, ]. ƒsd -, [, ]. ƒsd si, [-, ]. ƒsd si +, k [-, ]. Fid the orm of the rtitio P,.,.,.,.6, 6.. Fid the orm of the rtitio P -, -.6, -.,,.8, 6.. 8k. k imits of Uer Sums c. sk + d d. sk - d For the fuctios i Eercises fid formul for the uer sum otied dividig the itervl [, ] ito equl suitervls. The tke limit of these sums s : q to clculte the re uder the curve over [, ]. k k k k Evlute the sums i Eercises k. k c. k.. k. k c. k k k 7 k k k k. s - kd k. k. s - k d. sk - d k 6 k. ƒsd - over the itervl [, ]. 6. ƒsd over the itervl [, ]. 7. ƒsd + over the itervl [, ]. 8. ƒsd over the itervl [, ]. 9. ƒsd + over the itervl [, ].. ƒsd + over the itervl [, ]. 6 k Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle

23 AW/Thoms_ch-9 8// 9:7 AM Pge. The Defiite Itegrl. The Defiite Itegrl I Sectio. 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 rtitios of [, ] roches zero. For geerl Riem sums the suitervls of the rtitios eed ot hve equl widths. The limitig rocess 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 rtitios of [, ] roches zero, the vlues of the corresodig Riem Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle

24 AW/Thoms_ch-9 8// 9:7 AM Pge Chter : Itegrtio sums roch limitig vlue I. Wht we me this covergig ide is tht Riem sum will e close to the umer I rovided tht the orm of its rtitio is sufficietl smll (so tht ll of its suitervls hve thi eough widths). We itroduce the smol P s smll ositive umer tht secifies how close to I the Riem sum must e, d the smol d s secod smll ositive umer tht secifies how smll the orm of rtitio must e i order for tht to he. Here is recise 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 k ƒsck d k if the followig coditio is stisfied: Give umer P 7 there is corresodig umer d 7 such tht for ever rtitio P,, Á, 6 of [, ] with 7P7 6 d d choice of ck i [k -, k], we hve ` ƒsck d k - I ` 6 P. k eiiz itroduced ottio for the defiite itegrl tht ctures its costructio s limit of Riem sums. He evisioed the fiite sums g k ƒsck d k ecomig ifiite sum of fuctio vlues ƒ() multilied ifiitesiml suitervl widths d. The sum smol is relced i the limit the itegrl smol, whose origi is i the letter S. The fuctio vlues ƒsck d re relced cotiuous selectio of fuctio vlues ƒ(). The suitervl widths k ecome the differetil d. It is s if we re summig ll roducts of the form ƒsd # d s goes from to. While this ottio ctures the rocess of costructig itegrl, it is Riem s defiitio tht gives recise 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 resect to. The comoet rts i the itegrl smol lso hve mes: Uer limit of itegrtio Itegrl sig ower limit of itegrtio The fuctio is the itegrd. is the vrile of itegrtio. f() d Itegrl of f from to Whe ou fid the vlue of the itegrl, ou hve evluted the itegrl. Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle

25 AW/Thoms_ch-9 8// 9:7 AM Pge. The Defiite Itegrl 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 rtitio P with orm goig to zero, d m choices of oits ck for ech rtitio. 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 ƒsck d k I ƒ ƒ P ƒ ƒ : k ƒsd d. Whe ech rtitio hs equl suitervls, ech of width s - d>, we will lso write lim ƒsck d I : q k ƒsd d. The limit is lws tke s the orm of the rtitios roches zero d the umer of suitervls goes to ifiit. The vlue of the defiite itegrl of fuctio over rticulr itervl deeds o the fuctio, ot o the letter we choose to rereset its ideedet vrile. If we decide to use t or u isted of, we siml 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 ck so tht ƒsck d gives the mimum vlue of ƒ o [k -, k], givig uer sum. We c choose ck to give the miimum vlue of ƒ o [k -, k], givig lower sum. We c ick ck to e the midoit of [k -, k], the rightmost oit k, or rdom oit. We c tke the rtitios of equl or vrig widths. I ech cse we get the sme limit for g k ƒsck d k s 7 P7 :. The ide ehid Theorem is tht Riem sum ssocited with rtitio is o more th the uer sum of tht rtitio d o less th the lower sum. The uer d lower sums coverge to the sme vlue whe 7 P7 :. All other Riem sums lie etwee the uer d lower sums d hve the sme limit. A roof of Theorem ivolves creful lsis of fuctios, rtitios, d limits log this lie of thikig d is left to more dvced tet. A idictio of this roof is give i Eercises 8 d 8. Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle

26 AW/Thoms_ch-9 8// 9:7 AM Pge 6 6 Chter : Itegrtio Theorem ss othig out how to clculte defiite itegrls. A method of clcultio will e develoed i Sectio., through coectio to the rocess 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 iecewise-cotiuous fuctios defied i the Additiol Eercises t the ed of this chter. (The ltter re cotiuous ecet t fiite umer of oits i [, ].) For itegrilit to fil, fuctio eeds to e sufficietl discotiuous so tht the regio etwee its grh d the -is cot e roimted well icresigl thi rectgles. Here is emle of fuctio tht is ot itegrle. EXAMPE A Noitegrle Fuctio o [, ] The fuctio ƒsd e,, if is rtiol if is irrtiol 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 jums u d dow too errticll over [, ] to llow the regio eeth its grh d ove the -is to e roimted rectgles, o mtter how thi the re. We show, i fct, tht uer sum roimtios d lower sum roimtios coverge to differet limitig vlues. If we ick rtitio P of [, ] d choose ck to e the mimum vlue for ƒ o [k -, k] the the corresodig Riem sum is U ƒsck d k sd k, k k sice ech suitervl [k -, k] cotis rtiol umer where ƒsck d. Note tht the legths of the itervls i the rtitio sum to, g k k. So ech such Riem sum equls, d limit of Riem sums usig these choices equls. O the other hd, if we ick ck to e the miimum vlue for ƒ o [k -, k], the the Riem sum is ƒsck d k sd k, k k sice ech suitervl [k -, k] cotis irrtiol umer ck where ƒsck d. The limit of Riem sums usig these choices equls zero. Sice the limit deeds o the choices of ck, the fuctio ƒ is ot itegrle. Proerties of Defiite Itegrls I defiig ƒsd d s limit of sums g k ƒsck d k, we moved from left to right cross the itervl [, ]. Wht would he 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 ositive. With the sme choices of ck i ech suitervl, the sig of Riem sum would chge, s would the sig of the limit, the itegrl Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle

27 AW/Thoms_ch-9 8// 9:7 AM Pge 7. The Defiite Itegrl 7 ƒsd d. Sice we hve ot reviousl 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 ƒsck d k is zero whe the itervl width k, we defie ƒsd d. Theorem sttes seve roerties of itegrls, give s rules tht the stisf, icludig the two ove. These rules ecome ver useful i the rocess of comutig itegrls. We will refer to them reetedl to simlif our clcultios. Rules through 7 hve geometric iterrettios, show i Figure.. The grhs i these figures re of ositive fuctios, ut the rules l to geerl itegrle fuctios. THEOREM Whe ƒ d g re itegrle, the defiite itegrl stisfies Rules to 7 i Tle.. TABE. Rules stisfied defiite itegrls. Order of Itegrtio:. Zero Width Itervl:. Costt Multile: ƒsd d - ƒsd d A Defiitio ƒsd d Also Defiitio kƒsd d k ƒsd d -ƒsd d - ƒsd d. Sum d Differece:. Additivit: k - sƒsd ; gsdd d 6. A Numer k ƒsd d ; c ƒsd d + gsd d c ƒsd d ƒsd d M-Mi Iequlit: If ƒ hs mimum vlue m ƒ d miimum vlue mi ƒ o [, ], the mi ƒ # s - d ƒsd d m ƒ # s - d. 7. Domitio: ƒsd Ú gsd o [, ] Q ƒsd d Ú gsd d ƒsd Ú o [, ] Q ƒsd d Ú Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle (Secil Cse)

28 AW/Thoms_ch-9 8// 9:7 AM Pge 8 8 Chter : Itegrtio f () f () g() f () g() f () f () () Costt Multile: () Zero Width Itervl: ƒsd d. (Show for k.) (The re over oit is.) (c) Sum: kƒsd d k sƒsd + gsdd d ƒsd d. (Ares dd) ƒsd d + gsd d f () f () m f c f () d f () f () d mi f g() c (d) Additivit for defiite itegrls: ƒsd d + FIGURE. c ƒsd d (e) M-Mi Iequlit: c ƒsd d (f ) Domitio: mi ƒ # s - d ƒsd d m ƒ # s - d ƒsd Ú gsd o [, ] Q ƒsd d Ú gsd d While Rules d re defiitios, Rules to 7 of Tle. must e roved. The roofs re sed o the defiitio of the defiite itegrl s limit of Riem sums. The followig is roof of oe of these rules. Similr roofs c e give to verif the other roerties i Tle.. 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 rtitio of [, ] d for ever choice of the oits ck, mi ƒ # s - d mi ƒ # k k k - k mi ƒ # k Costt Multile Rule ƒsck d k mi ƒ ƒsck d m ƒ # k ƒsck d m f k k k m ƒ # k k m ƒ # s - d. Coright.Ö kqo ł _ s)j ææþ ÒÃ ê} ÛrNŁ }«"Ò»õã+sPîï Costt Multile Rule

29 AW/Thoms_ch-9 8// 9:7 AM Pge 9. The Defiite Itegrl 9 I short, ll Riem sums for ƒ o [, ] stisf the iequlit mi ƒ # s - d ƒsck d k m ƒ # s - d. k Hece their limit, the itegrl, does too. EXAMPE Usig the Rules for Defiite Itegrls Suose tht - ƒsd d, ƒsd d -, - hsd d 7. The. ƒsd d - ƒsd d - s -d. Rule - [ƒsd + hsd] d - ƒsd d + - hsd d Rules d sd + s7d. - ƒsd d EXAMPE - ƒsd d + ƒsd d + s -d Rule Fidig Bouds for Itegrl Show tht the vlue of + cos d is less th >. The M-Mi Iequlit for defiite itegrls (Rule 6) ss tht mi ƒ # s - d is lower oud for the vlue of ƒsd d d tht m ƒ # s - d is uer oud. Solutio The mimum vlue of + cos o [, ] is +, so + cos d # s - d. Sice + cos d is ouded from ove (which is. Á ), the itegrl is less th >. Are Uder the Grh of Noegtive Fuctio We ow mke recise the otio of the re of regio with curved oudr, cturig the ide of roimtig regio icresigl m rectgles. The re uder the grh 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. Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle

30 AW/Thoms_ch-9 8// 9:7 AM Pge Chter : Itegrtio For the first time we hve rigorous defiitio for the re of regio whose oudr is the grh of cotiuous fuctio. We ow l this to simle emle, the re uder stright lie, where we c verif tht our ew defiitio grees with our revious otio of re. EXAMPE Are Uder the ie FIGURE. The regio i Emle is trigle. Comute Solutio d d fid the re A uder over the itervl [, ], 7. The regio of iterest is trigle (Figure.). We comute the re i two ws. () To comute the defiite itegrl s the limit of Riem sums, we clculte lim ƒ ƒ P ƒ ƒ : g k ƒsck d k for rtitios whose orms go to zero. Theorem tells us tht it does ot mtter how we choose the rtitios or the oits ck s log s the orms roch zero. All choices give the ect sme limit. So we cosider the rtitio P tht sudivides the itervl [, ] ito suitervls of equl width s - d> >, d we choose ck to e the right edoit i ech suitervl. The rtitio is k P e,,,, Á, f d ck. So k k k # ƒsck d ƒsck d ck k k k k s + d # Costt Multile Rule Sum of First Itegers s + d As : q d 7P7 :, this lst eressio o the right hs the limit >. Therefore, 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 >. Emle c e geerlized to itegrte ƒsd over closed itervl [, ], 6 6. d d + d - Rule d + +. d Rule Emle Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle

31 AW/Thoms_ch-9 8// 9:7 AM Pge. The Defiite Itegrl I coclusio, we hve the followig rule for itegrtig f() : d 6 () This comuttio gives the re of trezoid (Figure.). 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 trezoid droig 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 grh d ove [, ] mius the re elow [, ] d over the grh. The followig results c lso e estlished usig Riem sum clcultio similr to tht i Emle (Eercises 7 d 76). -, FIGURE. The re of this trezoidl regio is A s - d>. c d cs - d, d c costt () 6 () -, Averge Vlue of Cotiuous Fuctio Revisited I Sectio. we itroduced iformll the verge vlue of oegtive cotiuous fuctio ƒ over itervl [, ], ledig us to defie this verge s the re uder the grh of ƒsd divided -. I itegrl ottio we write this s f () (ck, f (ck )) Averge ck FIGURE. A smle of vlues of fuctio o itervl [, ]. ƒsd d. - We c use this formul to give recise defiitio of the verge vlue of cotiuous (or itegrle) fuctio, whether ositive, 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 smle them i orderl w. We divide [, ] ito suitervls of equl width s - d> d evlute ƒ t oit ck i ech (Figure.). The verge of the smled vlues is ƒsc d + ƒsc d + Á + ƒsc d ƒsck d k ƒsck d - k ƒsck d - k Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle -, so -

32 AW/Thoms_ch-9 8// 9:7 AM Pge Chter : Itegrtio The verge is otied dividig Riem sum for ƒ o [, ] s - d. As we icrese the size of the smle d let the orm of the rtitio roch zero, the verge roches (>( - )) ƒsd d. Both oits 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 EXAMPE f () 兹 Fidig Averge Vlue Fid the verge vlue of ƒsd - o [-, ]. ƒsd d. - FIGURE. The verge vlue of ƒsd - o [ -, ] is > (Emle ). We recogize ƒsd - s fuctio whose grh is the uer semicircle of rdius cetered t the origi (Figure.). The re etwee the semicircle d the -is from - to c e comuted usig the geometr formul Solutio Are # r # sd. Becuse ƒ is oegtive, the re is lso the vlue of the itegrl of ƒ from - to, - - d. Therefore, the verge vlue of ƒ is vsƒd - d sd. - s -d - Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle

33 AW/Thoms_ch-9 8// 9:7 AM Pge Chter : Itegrtio EXERCISES. Eressig imits s Itegrls. lim k, where P is rtitio of [, ] ƒ ƒ P ƒ ƒ : k - ck 6. lim - ck k, where P is rtitio of [, ] ƒ ƒ P ƒ ƒ : Eress the limits i Eercises 8 s defiite itegrls... lim ck ƒ ƒpƒ ƒ : k k, where P is rtitio of [, ] lim ck k, where P is rtitio of [ -, ] lim ssec ck d k, where P is rtitio of [- >, ] ƒ ƒ P ƒ ƒ : k lim sck - ck d k, where P is rtitio of [-7, ] ƒ ƒpƒ ƒ : k. 7. ƒ ƒpƒ ƒ : k. k 8. lim st ck d k, where P is rtitio of [, >] ƒ ƒ P ƒ ƒ : k lim ck k, where P is rtitio of [, ] ƒ ƒpƒ ƒ : k Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle

34 AW/Thoms_ch-9 8// 9:7 AM Pge. The Defiite Itegrl Usig Proerties d Kow Vlues to Fid Other Itegrls 9. Suose tht ƒ d g re itegrle d tht ƒsd d -, ƒsd d 6, g sd d 8. Use the rules i Tle. to fid. g sd d. c. g sd d d. [ƒsd - g sd] d [ƒsd - g sd] d f. 9 ƒsd d, 7 Use the rules i Tle. to fid [ƒsd - hsd] d 7 d Suose tht ƒsd d. Fid.. Suose tht - g std dt. Fid. g std dt. g sud du [ - g sd] d - ƒszd d. ƒszd dz ƒstd dt.. Suose tht h is itegrle d tht - hsrd dr d - hsrd dr 6. Fid 8... u du. d 7. s ds /. d. d u du. t dt 9. 7 r dr d d 8. d - 7 d. d. d 8. st - d dt + z dz. u du sz - d dz 8. u du / 9. A t - B dt d s + - d d. s + - d d hsrd dr. - hsud du Usig Are to Evlute Defiite Itegrls I Eercises, grh the itegrds d use res to evlute the itegrls... d Use the rules i Tle. d Equtios () () to evlute the itegrls i Eercises 9.. dz 7. Fid. g srd - t dt, dr. Suose tht ƒ is itegrle d tht ƒszd dz d c Use the results of Equtios () d () to evlute the itegrls i Eercises 7 8. ƒszd dz [- ƒsd] d d. d, Evlutios 6. ƒstd dt c. 6. ƒsud du 6 6 / [hsd - ƒsd] d f.. s ds, 7 ƒsd d e. 7 ƒsd d A + - B d - d, 7. - [ƒsd + hsd] d 7 9 c. hsd d. 9 - ƒsd d. s - ƒ ƒ d d - 9 ƒsd d -, s - ƒ ƒ d d - Use res to evlute the itegrls i Eercises 6.. Suose tht ƒ d h re itegrle d tht d -. ƒsd d e. ƒ ƒ d d 9.. ƒsd d 7. + d - / 6. / s - + d d Fidig Are I Eercises use defiite itegrl to fid the re of the regio etwee the give curve d the -is o the itervl [, ] Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle

35 AW/Thoms_ch-9 8// 9:7 AM Pge Chter : Itegrtio Averge Vlue I Eercises 6, grh the fuctio d fid its verge vlue over the give itervl.. ƒsd - o C, D o [, ] 7. ƒsd ƒsd - o [, ] 6. ƒsd - 9. ƒstd st - d 6. ƒstd t - t 6. g sd ƒ ƒ - o. [-, ],. [, ], d c. [-, ] 6. hsd - ƒ ƒ o. [-, ],. [, ], d c. [-, ] 6. Wht vlues of d miimize the vlue of. vsƒ + gd vsƒd + vsgd s - d d? 6. Use the M-Mi Iequlit to fid uer d lower ouds for the vlue of d (Cotiutio of Eercise 6) Use the M-Mi Iequlit to fid uer d lower ouds for d + d d.. + Add these to rrive t imroved estimte of d Show tht the vlue of + 8 d lies etwee.8 d. [, ]. 76. Use limits of Riem sums s i Emle to estlish Equtio ().. Suose the grh of cotiuous fuctio ƒ() rises stedil s moves from left to right cross itervl [, ]. et P e rtitio of [, ] ito suitervls of legth s - d>. Show referrig to the ccomig figure tht the differece etwee the uer d lower sums for ƒ o this rtitio c e rereseted grhicll s the re of rectgle R whose dimesios re [ƒsd - ƒsd]. (Hit: The differece U - is the sum of res of rectgles whose digols Q Q, Q Q, Á, Q - Q lie log the curve. There is o overlig whe these rectgles re shifted horizotll oto R.). Suose tht isted of eig equl, the legths k of the suitervls of the rtitio of [, ] vr i size. Show tht where m is the orm of P, d hece tht lim ƒ ƒ P ƒ ƒ : su - d. 69. Itegrls of oegtive fuctios Use the M-Mi Iequlit to show tht if ƒ is itegrle the f () [, ] Q f() f() ƒsd d Ú. Q 7. Itegrls of oositive fuctios Show tht if ƒ is itegrle the Q o o U - ƒ ƒsd - ƒsd ƒ m, 67. Show tht the vlue of sis d d cot ossil e. ƒsd ƒsd g sd Do these rules ever hold? Give resos for our swers. o if s umer kd 77. Uer d lower sums for icresig fuctios ƒsd Ú ƒsd d? 7. Use limits of Riem sums s i Emle to estlish Equtio (). s - d d? (Hit: Where is the itegrd ositive?) 7. It would e ice if verge vlues of itegrle fuctios oeed the followig rules o itervl [, ]. c. vsƒd vsgd 6. Wht vlues of d mimize the vlue of. vsƒd d. vskƒd k vsƒd Theor d Emles Give resos for our swer. [-, ] 7. If v(ƒ) rell is ticl vlue of the itegrle fuctio ƒ() o [, ], the the umer v(ƒ) should hve the sme itegrl over [, ] tht ƒ does. Does it? Tht is, does o [, ] o [, ] o 7. The iequlit sec Ú + s >d holds o s - >, >d. Use it to fid lower oud for the vlue of sec d. [, ] Q R Q ƒsd d. 7. Use the iequlit si, which holds for Ú, to fid uer oud for the vlue of si d. Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle

36 AW/Thoms_ch-9 8// 9:7 AM Pge. The Defiite Itegrl 78. Uer d lower sums for decresig fuctios (Cotiutio of Eercise 77). Drw figure like the oe i Eercise 77 for cotiuous fuctio ƒ() whose vlues decrese stedil s moves from left to right cross the itervl [, ]. et P e rtitio of [, ] ito suitervls of equl legth. Fid eressio for U - tht is logous to the oe ou foud for U - i Eercise 77. f (). Suose tht isted of eig equl, the legths k of the suitervls of P vr i size. Show tht the iequlit U - ƒ ƒsd - ƒsd ƒ m k k of Eercise 77 still holds d hece tht lim ƒ ƒ P ƒ ƒ : su - d. 79. Use the formul si h + si h + si h + Á + si mh cos sh>d - cos ssm + s>ddhd si sh>d to fid the re uder the curve si from to > i two stes:. Prtitio the itervl [, >] ito suitervls of equl legth d clculte the corresodig uer sum U; the. Fid the limit of U s : q d s - d> :. k k k k 8. Suose tht ƒ is cotiuous d oegtive over [, ], s i the figure t the right. B isertig oits,, Á, k -, k, Á, - s show, divide [, ] ito suitervls of legths -, -, Á, - -, which eed ot e equl.. If mk mi ƒsd for i the k th suitervl6, eli the coectio etwee the lower sum m + m + Á + m d the shded regio i the first rt of the figure.. If Mk m ƒsd for i the k th suitervl6, eli the coectio etwee the uer sum U M + M + Á + M Plecher, The Mthemtics Techer, Vol. 8, No. 6,. 6, Setemer 99.) d the shded regio i the secod rt of the figure. c. Eli the coectio etwee U - d the shded regios log the curve i the third rt of the figure. 8. We s ƒ is uiforml cotiuous o [, ] if give P 7 there is d 7 such tht if, re i [, ] d ƒ - ƒ 6 d the ƒ ƒs d - ƒs d ƒ 6 P. It c e show tht cotiuous fuctio o [, ] is uiforml cotiuous. Use this d the figure t the right to show tht if ƒ is cotiuous d P 7 is give, it is ossile to mke U - P # s - d mkig the lrgest of the k s sufficietl smll. 8. If ou verge mi> h o -mi tri d the retur over the sme mi t the rte of mi> h, wht is our verge seed for the tri? Give resos for our swer. (Source: Dvid H. COMPUTER EXPORATIONS Fidig Riem Sums If our CAS c drw rectgles ssocited with Riem sums, use it to drw rectgles ssocited with Riem sums tht coverge to the itegrls i Eercises Use,,, d suitervls of equl legth i ech cse. 8. s - d d 8. Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle s + d d

37 AW/Thoms_ch-9 8// 9:7 AM Pge 6 6 Chter : Itegrtio 8. - / cos d 86. sec d d. Solve the equtio ƒsd sverge vlued for usig the verge vlue clculted i rt (c) for the rtitioig c. Comute the verge vlue of the fuctio vlues geerted i rt (). ƒ ƒ d d (The itegrl s vlue is out.69.) 89. ƒsd si Averge Vlue I Eercises 89 9, use CAS to erform the followig stes: 9. ƒsd si 9. ƒsd si. Plot the fuctios over the give itervl.. Prtitio the itervl ito,, d suitervls of equl legth, d evlute the fuctio t the midoit of ech suitervl. Co8 ëü;ł Z ¹ ³B` A³s7ñ 9QdãÒÝ8²&ž tcúžµ ôn À øñ!²jhhœ² 9z~ûÇ«å [, ] o 9. ƒsd si o o o [, ] c, d c, d

38 AW/Thoms_ch-9 8// 9:7 AM Pge 6 6 Chter : Itegrtio The Fudmetl Theorem of Clculus. I this sectio we reset the Fudmetl Theorem of Clculus, which is the cetrl theorem of itegrl clculus. It coects itegrtio d differetitio, elig us to comute itegrls usig tiderivtive of the itegrd fuctio rther th tkig limits of Riem sums s we did i Sectio.. eiiz d Newto eloited this reltioshi d strted mthemticl develomets tht fueled the scietific revolutio for the et ers. Alog the w, we reset the itegrl versio of the Me Vlue Theorem, which is other imortt theorem of itegrl clculus d used to rove the Fudmetl Theorem. HISTORICA BIOGRAPHY Sir Isc Newto (6 77) Me Vlue Theorem for Defiite Itegrls f () f (c), verge height c FIGURE.6 The vlue ƒ(c) i the Me Vlue Theorem is, i sese, the verge (or me) height of ƒ o [, ]. Whe ƒ Ú, the re of the rectgle is the re uder the grh of ƒ from to, ƒscds - d I the revious sectio, we defied the verge vlue of cotiuous fuctio over closed itervl [, ] s the defiite itegrl ƒsd d divided the legth or width - of the itervl. The Me Vlue Theorem for Defiite Itegrls sserts tht this verge vlue is lws tke o t lest oce the fuctio ƒ i the itervl. The grh i Figure.6 shows ositive cotiuous fuctio ƒsd defied over the itervl [, ]. Geometricll, the Me Vlue Theorem ss tht there is umer c i [, ] such tht the rectgle with height equl to the verge vlue ƒ(c) of the fuctio d se width - hs ectl the sme re s the regio eeth the grh of ƒ from to. THEOREM The Me Vlue Theorem for Defiite Itegrls If ƒ is cotiuous o [, ], the t some oit c i [, ], ƒscd ƒsd d. - ƒsd d. Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle

39 AW/Thoms_ch-9 8// 9:7 AM Pge 7. The Fudmetl Theorem of Clculus 7 Proof If we divide oth sides of the M-Mi Iequlit (Tle., Rule 6) s - d, we oti mi ƒ Sice ƒ is cotiuous, the Itermedite Vlue Theorem for Cotiuous Fuctios (Sectio.6) ss tht ƒ must ssume ever vlue etwee mi ƒ d m ƒ. It must therefore s sume the vlue s>s - dd ƒsd d t some oit c i [, ]. f () Averge vlue / ot ssumed The cotiuit of ƒ is imortt here. It is ossile tht discotiuous fuctio ever equls its verge vlue (Figure.7). ƒsd d m ƒ. - FIGURE.7 A discotiuous fuctio eed ot ssume its verge vlue. EXAMPE Alig the Me Vlue Theorem for Itegrls Fid the verge vlue of ƒsd - o [, ] d where ƒ ctull tkes o this vlue t some oit i the give domi. Solutio ƒsd d - s - d d d d - s - d - vsƒd Sectio., Eqs. () d () FIGURE.8 The re of the rectgle with se [, ] d height > (the verge vlue of the fuctio ƒsd - ) is equl to the re etwee the grh of ƒ d the -is from to (Emle ).. The verge vlue of ƒsd - over [, ] is >. The fuctio ssumes this vlue whe - > or >. (Figure.8) I Emle, we ctull foud oit c where ƒ ssumed its verge vlue settig ƒ() equl to the clculted verge vlue d solvig for. It s ot lws ossile to solve esil for the vlue c. Wht else c we ler from the Me Vlue Theorem for itegrls? Here s emle. EXAMPE Show tht if ƒ is cotiuous o [, ], Z, d if ƒsd d, the ƒsd t lest oce i [, ]. Solutio The verge vlue of ƒ o [, ] is vsƒd ƒsd d - - #. B the Me Vlue Theorem, ƒ ssumes this vlue t some oit c H [, ]. Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle

40 AW/Thoms_ch-9 8// 9:7 AM Pge 8 8 Chter : Itegrtio Fudmetl Theorem, Prt If ƒ(t) is itegrle fuctio over fiite itervl I, the the itegrl from fied umer H I to other umer H I defies ew fuctio F whose vlue t is Fsd re F() f (t) t ƒstd dt. () For emle, if ƒ is oegtive d lies to the right of, the F() is the re uder the grh from to (Figure.9). The vrile is the uer limit of itegrtio of itegrl, ut F is just like other rel-vlued fuctio of rel vrile. For ech vlue of the iut, there is well-defied umericl outut, i this cse the defiite itegrl of ƒ from to. Equtio () gives w to defie ew fuctios, ut its imortce ow is the coectio it mkes etwee itegrls d derivtives. If ƒ is cotiuous fuctio, the the Fudmetl Theorem sserts tht F is differetile fuctio of whose derivtive is ƒ itself. At ever vlue of, FIGURE.9 The fuctio F() defied Equtio () gives the re uder the grh of ƒ from to whe ƒ is oegtive d 7. d d Fsd ƒstd dt ƒsd. d d To gi some isight ito wh this result holds, we look t the geometr ehid it. If ƒ Ú o [, ], the the comuttio of F sd from the defiitio of the derivtive mes tkig the limit s h : of the differece quotiet Fs + hd - Fsd. h For h 7, the umertor is otied sutrctig two res, so it is the re uder the grh of ƒ from to + h (Figure.). If h is smll, this re is roimtel equl to the re of the rectgle of height ƒ() d width h, which c e see from Figure.. Tht is, f (t) f () h Fs + hd - Fsd hƒsd. t FIGURE. I Equtio (), F() is the re to the left of. Also, Fs + hd is the re to the left of + h. The differece quotiet [Fs + hd - Fsd]>h is the roimtel equl to ƒ(), the height of the rectgle show here. Dividig oth sides of this roimtio h d lettig h :, it is resole to eect tht F sd lim h: Fs + hd - Fsd ƒsd. h This result is true eve if the fuctio ƒ is ot ositive, d it forms the first rt of the Fudmetl Theorem of Clculus. THEOREM The Fudmetl Theorem of Clculus Prt If ƒ is cotiuous o [, ] the Fsd ƒstd dt is cotiuous o [, ] d differetile o (, ) d its derivtive is ƒsd ; F sd d ƒstd dt ƒsd. d Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle ()

41 AW/Thoms_ch-9 8// 9:7 AM Pge 9. The Fudmetl Theorem of Clculus 9 Before rovig Theorem, we look t severl emles to gi etter uderstdig of wht it ss. EXAMPE Alig the Fudmetl Theorem Use the Fudmetl Theorem to fid () d cos t dt d () d dt d + t (c) d d if d (d) d if t si t dt cos t dt Solutio () d cos t dt cos d Eq. with ƒ(t) cos t d dt Eq. with ƒstd d + t + t + (c) Rule for itegrls i Tle. of Sectio. sets this u for the Fudmetl Theorem. () d d d t si t dt - t si t dt d d d d t si t dt d - si Rule (d) The uer limit of itegrtio is ot ut. This mkes comosite of the two fuctios, u cos t dt u. d We must therefore l the Chi Rule whe fidig d>d. d d d du # du d u d cos t dt du cos u # # du d du d coss d # cos Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle

42 AW/Thoms_ch-9 8// 9:7 AM Pge 6 6 Chter : Itegrtio EXAMPE Costructig Fuctio with Give Derivtive d Vlue Fid fuctio ƒsd o the domi s - >, >d with derivtive d t d tht stisfies the coditio ƒsd. Solutio The Fudmetl Theorem mkes it es to costruct fuctio with derivtive t tht equls t : t t dt. Sice sd t t dt, we hve ol to dd to this fuctio to costruct oe with derivtive t whose vlue t is : ƒsd t t dt +. Although the solutio to the rolem i Emle stisfies the two required coditios, ou might sk whether it is i useful form. The swer is es, sice tod we hve comuters d clcultors tht re cle of roimtig itegrls. I Chter 7 we will ler to write the solutio i Emle ectl s cos l ` cos ` +. We ow give roof of the Fudmetl Theorem for ritrr cotiuous fuctio. Proof of Theorem We rove the Fudmetl Theorem lig the defiitio of the derivtive directl to the fuctio F(), whe d + h re i (, ). This mes writig out the differece quotiet Fs + hd - Fsd h () d showig tht its limit s h : is the umer ƒ() for ech i (, ). Whe we relce Fs + hd d F() their defiig itegrls, the umertor i Equtio () ecomes +h Fs + hd - Fsd ƒstd dt - ƒstd dt. The Additivit Rule for itegrls (Tle., Rule ) simlifies the right side to +h ƒstd dt, so tht Equtio () ecomes Fs + hd - Fsd [Fs + hd - Fsd] h h h +h ƒstd dt. Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle ()

43 AW/Thoms_ch-9 8// 9:7 AM Pge 6. The Fudmetl Theorem of Clculus 6 Accordig to the Me Vlue Theorem for Defiite Itegrls, the vlue of the lst eressio i Equtio () is oe of the vlues tke o ƒ i the itervl etwee d + h. Tht is, for some umer c i this itervl, h +h ƒstd dt ƒscd. () As h :, + h roches, forcig c to roch lso (ecuse c is tred etwee d + h). Sice ƒ is cotiuous t, ƒ(c) roches ƒ(): lim ƒscd ƒsd. (6) h: Goig ck to the egiig, the, we hve Fs + hd - Fsd df lim d h h: h: h Defiitio of derivtive +h lim ƒstd dt Eq. () lim ƒscd Eq. () ƒsd. Eq. (6) h: If or, the the limit of Equtio () is iterreted s oe-sided limit with h : + or h : -, resectivel. The Theorem i Sectio. shows tht F is cotiuous for ever oit of [, ]. This cocludes the roof. Fudmetl Theorem, Prt (The Evlutio Theorem) We ow come to the secod rt of the Fudmetl Theorem of Clculus. This rt descries how to evlute defiite itegrls without hvig to clculte limits of Riem sums. Isted we fid d evlute tiderivtive t the uer d lower limits of itegrtio. THEOREM (Cotiued) The Fudmetl Theorem of Clculus Prt If ƒ is cotiuous t ever oit of [, ] d F is tiderivtive of ƒ o [, ], the ƒsd d Fsd - Fsd. Proof Prt of the Fudmetl Theorem tells us tht tiderivtive of ƒ eists, mel Gsd ƒstd dt. Thus, if F is tiderivtive of ƒ, the Fsd Gsd + C for some costt C for 6 6 ( Corollr of the Me Vlue Theorem for Derivtives, Sectio.). Sice oth F d G re cotiuous o [, ], we see tht F() G() + C lso holds whe d tkig oe-sided limits (s : + d : - d. Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle

44 AW/Thoms_ch-9 8// 9:7 AM Pge 6 6 Chter : Itegrtio Evlutig Fsd - Fsd, we hve Fsd - Fsd [Gsd + C] - [Gsd + C] Gsd - Gsd ƒstd dt - ƒstd dt ƒstd dt - ƒstd dt. The theorem ss tht to clculte the defiite itegrl of ƒ over [, ] ll we eed to do is:.. Fid tiderivtive F of ƒ, d Clculte the umer ƒsd d Fsd - Fsd. The usul ottio for Fsd - Fsd is Fsd d or cfsd d, deedig o whether F hs oe or more terms. EXAMPE () cos d si d () Evlutig Itegrls -> si - si - sec t d sec d -/ - (c) sec - sec - - d c / + d csd/ + d - csd/ + d [8 + ] - []. The rocess used i Emle ws much esier th Riem sum comuttio. The coclusios of the Fudmetl Theorem tell us severl thigs. Equtio () c e rewritte s df d ƒsd, ƒstd dt d d which ss tht if ou first itegrte the fuctio ƒ d the differetite the result, ou get the fuctio ƒ ck gi. ikewise, the equtio df dt ƒstd dt Fsd - Fsd dt ss tht if ou first differetite the fuctio F d the itegrte the result, ou get the fuctio F ck (djusted itegrtio costt). I sese, the rocesses of itegrcoright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle

45 AW/Thoms_ch-9 8// 9:7 AM Pge 6. The Fudmetl Theorem of Clculus 6 tio d differetitio re iverses of ech other. The Fudmetl Theorem lso ss tht ever cotiuous fuctio ƒ hs tiderivtive F. Ad it ss tht the differetil equtio d>d ƒsd hs solutio (mel, the fuctio F()) for ever cotiuous fuctio ƒ. Totl Are The Riem sum cotis terms such s ƒsck d k which give the re of rectgle whe ƒsck d is ositive. Whe ƒsck d is egtive, the the roduct ƒsck d k is the egtive of the rectgle s re. Whe we dd u such terms for egtive fuctio we get the egtive of the re etwee the curve d the -is. If we the tke the solute vlue, we oti the correct ositive re. EXAMPE 6 Fidig Are Usig Atiderivtives Clculte the re ouded the -is d the rol Solutio We fid where the curve crosses the -is settig s + ds - d, which gives Are The curve i Figure. is rch of rol, d it is iterestig to ote tht the re uder such rch is ectl equl to two-thirds the se times the ltitude: sd 6. 6 si Are d s6 - - d d c6 - FIGURE. The re of this rolic rch is clculted with defiite itegrl (Emle 6).. The curve is sketched i Figure., d is oegtive o [-, ]. The re is or To comute the re of the regio ouded the grh of fuctio ƒsd d the -is requires more cre whe the fuctio tkes o oth ositive d egtive vlues. We must e creful to rek u the itervl [, ] ito suitervls o which the fuctio does t chge sig. Otherwise we might get ccelltio etwee ositive d egtive siged res, ledig to icorrect totl. The correct totl re is otied ddig the solute vlue of the defiite itegrl over ech suitervl where ƒ() does ot chge sig. The term re will e tke to me totl re. EXAMPE 7 FIGURE. The totl re etwee si d the -is for is the sum of the solute vlues of two itegrls (Emle 7). Ccelig Ares Figure. shows the grh of the fuctio ƒsd si etwee d. Comute () the defiite itegrl of ƒ() over [, ]. () the re etwee the grh of ƒ() d the -is over [, ]. Coright Perso Eductio, Ic., ulishig s Perso AŁ Ë+Ç è`mxûä

46 AW/Thoms_ch-9 8// 9:7 AM Pge 6 6 Chter : Itegrtio The defiite itegrl for ƒsd si is give Solutio si d - cos d - [cos - cos ] - [ - ]. The defiite itegrl is zero ecuse the ortios of the grh ove d elow the -is mke ccelig cotriutios. The re etwee the grh of ƒ() d the -is over [, ] is clculted rekig u the domi of si ito two ieces: the itervl [, ] over which it is oegtive d the itervl [, ] over which it is oositive. si d - cos d si d - cos d - [cos - cos ] - [- - ]. - [cos - cos ] - [ - s -d] -. The secod itegrl gives egtive vlue. The re etwee the grh d the is is otied ddig the solute vlues Are ƒ ƒ + ƒ - ƒ. Summr: To fid the re etwee the grh of ƒsd d the -is over the itervl [, ], do the followig:... Are 8 Are 8 Sudivide [, ] t the zeros of ƒ. Itegrte ƒ over ech suitervl. Add the solute vlues of the itegrls. EXAMPE 8 Fidig Are Usig Atiderivtives Fid the re of the regio etwee the -is d the grh of ƒ() - -, -. First fid the zeros of ƒ. Sice Solutio ƒsd - - s - - d s + ds - d, FIGURE. The regio etwee the curve - - d the -is (Emle 8). the zeros re, -, d (Figure.). The zeros sudivide [-, ] ito two suitervls: [-, ], o which ƒ Ú, d [, ], o which ƒ. We itegrte ƒ over ech suitervl d dd the solute vlues of the clculted itegrls. - s - - d d c - d - c + - d - s - - d d c d c - - d - The totl eclosed re is otied ddig the solute vlues of the clculted itegrls, Totl eclosed re `- `. Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle

47 AW/Thoms_ch-9 8// 9:7 AM Pge 6 6. The Fudmetl Theorem of Clculus EXERCISES. Evlutig Itegrls si.. - s + d d.. A + B d 6. - d 8. si d. sec d. csc u cot u du >. > + cos t dt > 6. -> 8. -> sr + d dr... - sec t +. dt t - cos st + dst + d dt u - du u. - d > s + s ds s. - u 9. du ƒ ƒ d 6 Derivtives of Itegrls Fid the derivtives i Eercises 7. t 9. cos t dt d d si 8. d du t u. d u du dt. sec t sec d + t dt. si st d dt. dt, t. t dt Fid d>d i Eercises 兹. differetitig the itegrl directl. d d 6. evlutig the itegrl d differetitig the result. 7. si u scos + ƒ cos ƒ d d Fid the res of the shded regios i Eercises > -, - cos t dt -> s8 + si d d ->. >, sec u t u du > 7.. -, >. csc d > , >6 >. 8. -, s + cos d d - > , d - Are I Eercises 7, fid the totl re etwee the regio d the -is. > d 9. s - + d d - -6> d ƒƒ 6, d. -. dt - t dt 6. + t t Evlute the itegrls i Eercises 6. 兹 sec t 7 cos t dt Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle t t

48 AW/Thoms_ch-9 8// 9:7 AM Pge Chter : Itegrtio Iitil Vlue Prolems Drwig Coclusios Aout Motio from Grhs Ech of the followig fuctios solves oe of the iitil vlue rolems i Eercises 7. Which fuctio solves which rolem? Give rief resos for our swers. 9. Suose tht ƒ is the differetile fuctio show i the ccomig grh d tht the ositio t time t (sec) of rticle movig log coordite is is. dt - t. c. - d, 7. d t sec t dt + s ƒsd d d. sec t dt + 8. sec, sd - 9. sec, dt - t., sd s -d sd - Eress the solutios of the iitil vlue rolems i Eercises i terms of itegrls. d sec, sd. d d +,. d meters. Use the grh to swer the followig questios. Give resos for our swers. sd - f () (, ) (, ) (, ) (, ) ds ƒstd,. dt sst d s d g std,. dt st d. Wht is the rticle s velocit t time t?. Is the ccelertio of the rticle t time t ositive, or egtive? c. Wht is the rticle s ositio t time t? Alictios. Archimedes re formul for rols Archimedes (87 B.C.), ivetor, militr egieer, hsicist, d the gretest mthemtici of clssicl times i the Wester world, discovered tht the re uder rolic rch is two-thirds the se times the height. Sketch the rolic rch h - sh> d, -> >, ssumig tht h d re ositive. The use clculus to fid the re of the regio eclosed etwee the rch d the -is. 6. Reveue from mrgil reveue Suose tht com s mrgil reveue from the mufcture d sle of egg eters is d. At wht time durig the first 9 sec does s hve its lrgest vlue? e. Aroimtel whe is the ccelertio zero? f. Whe is the rticle movig towrd the origi? w from the origi? g. O which side of the origi does the rticle lie t time t 9? 6. Suose tht g is the differetile fuctio grhed here d tht the ositio t time t (sec) of rticle movig log coordite is is t dr - >s + d, d where r is mesured i thousds of dollrs d i thousds of uits. How much moe should the com eect from roductio ru of thousd egg eters? To fid out, itegrte the mrgil reveue from to. 7. Cost from mrgil cost The mrgil cost of ritig oster whe osters hve ee rited is s 8 (7, 6.) (6, 6) 6 g() dollrs. Fid csd - csd, the cost of ritig osters. 8. (Cotiutio of Eercise 7.) Fid csd - csd, the cost of ritig osters. g sd d meters. Use the grh to swer the followig questios. Give resos for our swers. dc d 6 6 Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle 9

49 AW/Thoms_ch-9 8// 9:7 AM Pge 67. The Fudmetl Theorem of Clculus. Wht is the rticle s velocit t t? Give resos for our swers.. Is the ccelertio t time t ositive, or egtive?. h is twice-differetile fuctio of. c. Wht is the rticle s ositio t time t?. h d dh>d re oth cotiuous. d. Whe does the rticle ss through the origi? c. The grh of h hs horizotl tget t. e. Whe is the ccelertio zero? d. h hs locl mimum t. f. Whe is the rticle movig w from the origi? towrd the origi? e. h hs locl miimum t. g. O which side of the origi does the rticle lie t t 9? f. The grh of h hs iflectio oit t. g. The grh of dh>d crosses the -is t. T 69. The Fudmetl Theorem Theor d Emles 6. Show tht if k is ositive costt, the the re etwee the -is d oe rch of the curve si k is >k. 6. Fid lim : 6. Suose t dt. t + ƒstd dt - +. Fid ƒ(). 6. Fid ƒ() if ƒstd dt cos. 6. Fid the lieriztio of + ƒsd - 9 dt + t t. 66. Fid the lieriztio of lim h: If ƒ is cotiuous, we eect h +h ƒstd dt to equl ƒ(), s i the roof of Prt of the Fudmetl Theorem. For istce, if ƒstd cos t, the h +h cos t dt si s + hd - si. h (7) The right-hd side of Equtio (7) is the differece quotiet for the derivtive of the sie, d we eect its limit s h : to e cos. Grh cos for -. The, i differet color if ossile, grh the right-hd side of Equtio (7) s fuctio of for h,,., d.. Wtch how the ltter curves coverge to the grh of the cosie s h :. T 7. Reet Eercise 69 for ƒstd t. Wht is g sd + 67 sec st - d dt lim h: t Suose tht ƒ hs ositive derivtive for ll vlues of d tht ƒsd. Which of the followig sttemets must e true of the fuctio h s + hd -? h h: +h t dt lim Grh ƒsd for -. The grh the quotiet ss + hd - d>h s fuctio of for h,.,., d.. Wtch how the ltter curves coverge to the grh of s h :. g sd ƒstd dt? Give resos for our swers.. g is differetile fuctio of.. g is cotiuous fuctio of. c. The grh of g hs horizotl tget t. d. g hs locl mimum t. e. g hs locl miimum t. f. The grh of g hs iflectio oit t. g. The grh of dg>d crosses the -is t. 68. Suose tht ƒ hs egtive derivtive for ll vlues of d tht ƒsd. Which of the followig sttemets must e true of the fuctio hsd COMPUTER EXPORATIONS ƒstd dt? I Eercises 7 7, let Fsd ƒstd dt for the secified fuctio ƒ d itervl [, ]. Use CAS to erform the followig stes d swer the questios osed.. Plot the fuctios ƒ d F together over [, ].. Solve the equtio F sd. Wht c ou see to e true out the grhs of ƒ d F t oits where F sd? Is our oservtio ore out Prt of the Fudmetl Theorem couled with iformtio rovided the first derivtive? Eli our swer. c. Over wht itervls (roimtel) is the fuctio F icresig d decresig? Wht is true out ƒ over those itervls? d. Clculte the derivtive ƒ d lot it together with F. Wht c ou see to e true out the grh of F t oits where ƒ sd? Is our oservtio ore out Prt of the Fudmetl Theorem? Eli our swer. Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle

50 AW/Thoms_ch-9 8// 9:7 AM Pge Chter : Itegrtio 7. ƒsd - +, [, ] 9 7. ƒsd , c, d 7. ƒsd si cos, 7. ƒsd cos, d. Usig the iformtio from rts () (c), drw rough hdsketch of Fsd over its domi. The grh F() o our CAS to suort our sketch. 7., [, ] [, ] u() I Eercises 7 78, let Fsd ƒstd dt for the secified, u, d ƒ. Use CAS to erform the followig stes d swer the questios osed.. Fid the domi of F.. Clculte F sd d determie its zeros. For wht oits i its domi is F icresig? decresig? c. Clculte F sd d determie its zero. Idetif the locl etrem d the oits of iflectio of F. usd, ƒsd - ƒsd - 76., usd, 77., usd -, 78., usd -, ƒsd - - ƒsd - - I Eercises 79 d 8, ssume tht f is cotiuous d u() is twicedifferetile. usd 79. Clculte d d 8. Clculte d d ƒstd dt d check our swer usig CAS. usd ƒstd dt d check our swer usig CAS. Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle

51 AW/Thoms_ch-9 8// 9:7 AM Pge Chter : Itegrtio. Idefiite Itegrls d the Sustitutio Rule A defiite itegrl is umer defied tkig the limit of Riem sums ssocited with rtitios of fiite closed itervl whose orms go to zero. The Fudmetl Theorem of Clculus ss tht defiite itegrl of cotiuous fuctio c e comuted esil if we c fid tiderivtive of the fuctio. Atiderivtives geerll tur out to e more difficult to fid th derivtives. However, it is well worth the effort to ler techiques for comutig them. Recll from Sectio.8 tht the set of ll tiderivtives of the fuctio ƒ is clled the idefiite itegrl of ƒ with resect to, d is smolized ƒsd d. The coectio etwee tiderivtives d the defiite itegrl stted i the Fudmetl Theorem ow elis this ottio. Whe fidig the idefiite itegrl of fuctio ƒ, rememer tht it lws icludes ritrr costt C. We must distiguish crefull etwee defiite d idefiite itegrls. A defiite i tegrl ƒsd d is umer. A idefiite itegrl ƒsd d is fuctio lus ritrr costt C. So fr, we hve ol ee le to fid tiderivtives of fuctios tht re clerl recogizle s derivtives. I this sectio we egi to develo more geerl techiques for fidig tiderivtives. The first itegrtio techiques we develo re otied ivertig rules for fidig derivtives, such s the Power Rule d the Chi Rule. The Power Rule i Itegrl Form If u is differetile fuctio of d is rtiol umer differet from -, the Chi Rule tells us tht u+ du d. u d + d Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle

52 AW/Thoms_ch-9 8// 9:7 AM Pge Idefiite Itegrls d the Sustitutio Rule From other oit of view, this sme equtio ss tht u + >s + d is oe of the tiderivtives of the fuctio u sdu>dd. Therefore, u u+ du d + C. + d The itegrl o the left-hd side of this equtio is usull writte i the simler differetil form, u du, otied tretig the d s s differetils tht ccel. We re thus led to the followig rule. If u is differetile fuctio, the u du u+ + C + s Z -, rtiold. () Equtio () ctull holds for rel eoet Z -, s we see i Chter 7. I derivig Equtio (), we ssumed u to e differetile fuctio of the vrile, ut the me of the vrile does ot mtter d does ot er i the fil formul. We could hve rereseted the vrile with u, t,, or other letter. Equtio () ss tht wheever we c cst itegrl i the form u du, s Z - d, with u differetile fuctio d du its differetil, we c evlute the itegrl s [u + >s + d] + C. EXAMPE Usig the Power Rule + # d u # du d d et u +, du>d u > du u s>d + + C s>d + Itegrte, usig Eq. () with >. > u + C Simler form s + d> + C Relce u +. Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle

53 AW/Thoms_ch-9 8// 9:7 AM Pge 7 7 Chter : Itegrtio EXAMPE Adjustig the Itegrd Costt # t - # dt du u # dt dt u > du # u > + C > t - dt et u t -, du>dt. With the > out frot, the itegrl is ow i stdrd form. Itegrte, usig Eq. () with >. > u + C 6 Simler form st - d> + C 6 Relce u t -. Sustitutio: Ruig the Chi Rule Bckwrds The sustitutios i Emles d re istces of the followig geerl rule. THEOREM The Sustitutio Rule If u gsd is differetile fuctio whose rge is itervl I d ƒ is cotiuous o I, the ƒsgsddg sd d ƒsud du. Proof The rule is true ecuse, the Chi Rule, F(g()) is tiderivtive of ƒsgsdd # g sd wheever F is tiderivtive of ƒ: d Fsgsdd F sgsdd # g sd d ƒs gsdd # g sd. Chi Rule Becuse F ƒ If we mke the sustitutio u gsd the d Fsgsdd d d Fsgsdd + C Fsud + C ƒsgsddg sd d Fudmetl Theorem u gsd F sud du Fudmetl Theorem ƒsud du F ƒ Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle

54 AW/Thoms_ch-9 8// 9:7 AM Pge 7. Idefiite Itegrls d the Sustitutio Rule 7 The Sustitutio Rule rovides the followig method to evlute the itegrl ƒsgsddg sd d, whe ƒ d g re cotiuous fuctios:. Sustitute u gsd d du g sd d to oti the itegrl ƒsud du.. Itegrte with resect to u.. Relce u g() i the result. EXAMPE Usig Sustitutio cos u # cos s7u + d du du 7 cos u du 7 si u + C 7 si s7u + d + C 7 et u 7u +, du 7 du, s>7d du du. With the (>7) out frot, the itegrl is ow i stdrd form. Itegrte with resect to u, Tle.. Relce u 7u +. We c verif this solutio differetitig d checkig tht we oti the origil fuctio cos s7u + d. EXAMPE Usig Sustitutio si s d d si s d # d si u # du si u du s - cos ud + C - cos s d + C Coright Perso Eductio, Ic., ulishig s Pe ÇéÆðŁ Ž ܃}å et u, du d, s>d du d. Itegrte with resect to u. Relce u.

55 AW/Thoms_ch-9 8// 9:7 AM Pge 7 7 Chter : Itegrtio EXAMPE Usig Idetities d Sustitutio d sec d cos sec u # du sec u du t u + C t + C sec cos u, du d, d s>d du d t u sec u du u The success of the sustitutio method deeds o fidig sustitutio tht chges itegrl we cot evlute directl ito oe tht we c. If the first sustitutio fils, tr to simlif the itegrd further with dditiol sustitutio or two (see Eercises 9 d ). Altertivel, we c strt fresh. There c e more th oe good w to strt, s i the et emle. EXAMPE 6 Usig Differet Sustitutios Evlute z dz. z + Solutio We c use the sustitutio method of itegrtio s elortor tool: Sustitute for the most troulesome rt of the itegrd d see how thigs work out. For the itegrl here, we might tr u z + or we might eve ress our luck d tke u to e the etire cue root. Here is wht hes i ech cse. Solutio : Sustitute u z +. z dz z + du u > u -> du u > + C > > u + C sz + d> + C et u z +, du z dz. I the form u du Itegrte with resect to u. Relce u z +. Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle

56 AW/Thoms_ch-9 8// 9:7 AM Pge 7. Idefiite Itegrls d the Sustitutio Rule 7 Solutio : Sustitute u z + isted. z dz z + # z +, et u u z +, u du z dz. u du u u du u + C Itegrte with resect to u. sz + d> + C Relce u sz + d>. The Itegrls of si d cos Sometimes we c use trigoometric idetities to trsform itegrls we do ot kow how to evlute ito oes we c usig the sustitutio rule. Here is emle givig the itegrl formuls for si d cos which rise frequetl i lictios. EXAMPE 7 () () - cos - cos si d s - cos d d d cos d si si + C + C si d + cos d si + C + cos d + cos As i rt (), ut with sig chge Are Beeth the Curve si EXAMPE 8 si cos Figure. shows the grh of gsd si over the itervl [, ]. Fid () the defiite itegrl of g sd over [, ]. () the re etwee the grh of the fuctio d the -is over [, ]. Solutio () From Emle 7(), the defiite itegrl is FIGURE. The re eeth the curve si over [, ] equls squre uits (Emle 8). si d c si si si d c d - c d [ - ] - [ - ]. () The fuctio si is oegtive, so the re is equl to the defiite itegrl, or. Coright Perso EductioQ šîºïłuu Ù%ï zl> õúœ :6M«7ßPþ Ör

57 AW/Thoms_ch-9 8// 9:7 AM Pge 7 7 Chter : Itegrtio EXAMPE 9 V Vm V Vm si t Household Electricit We c model the voltge i our home wirig with the sie fuctio V Vv m V Vm si t, 6 t which eresses the voltge V i volts s fuctio of time t i secods. The fuctio rus through 6 ccles ech secod (its frequec is 6 hertz, or 6 Hz). The ositive costt Vm ( vee m ) is the ek voltge. The verge vlue of V over the hlf-ccle from to > sec (see Figure.) is > Vm si t dt s>d - > Vm ccos t d Vv FIGURE. The grh of the voltge V Vm si t over full ccle. Its verge vlue over hlf-ccle is Vm>. Its verge vlue over full ccle is zero (Emle 9). Vm [- cos + cos ] Vm. The verge vlue of the voltge over full ccle is zero, s we c see from Figure.. (Also see Eercise 6.) If we mesured the voltge with stdrd movig-coil glvometer, the meter would red zero. To mesure the voltge effectivel, we use istrumet tht mesures the squre root of the verge vlue of the squre of the voltge, mel Vrms sv dv. The suscrit rms (red the letters sertel) stds for root me squre. Sice the verge vlue of V svm d si t over ccle is sv dv >6 svm d, svm d si t dt s>6d - (Eercise 6, rt c), the rms voltge is Vrms svm d Vm. B The vlues give for household currets d voltges re lws rms vlues. Thus, volts c mes tht the rms voltge is. The ek voltge, otied from the lst equtio, is Vm Vrms # 6 volts, which is cosiderl higher. Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle

58 AW/Thoms_ch-9 8// 9:7 AM Pge 7 7 Chter : Itegrtio EXERCISES. Evlutig Itegrls Evlute the idefiite itegrls i Eercises usig the give sustitutios to reduce the itegrls to stdrd form.. sec t t t dt, u t. si d, u. si s d d, u. t t - cos si dt, u - cos Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle t

59 AW/Thoms_ch-9 8// 9:7 AM Pge 7. Idefiite Itegrls d the Sustitutio Rule. 6. 8s7 - d- d, s - d d, 9r dr 7. - r cos d,. csc u cot u du u + + u > - u -. Usig u cot u. Usig u csc u d + 8. Usig u + 8. Usig u + 8 Evlute the itegrls i Eercises s ds. ds s + 6. u - u du 8.. d s + d.. cos sz + d dz.. sec s + d d 6. si cos d 7. r r - dr cos u u si u 9. ss + s - s + dss + s - d ds su - u + 8u - dsu - u + d du t s + t d dt 6. + d 8. - d A 8 t sec d s + t d. u t, followed u, the w +. + si s - d si s - d cos s - d d. u -, followed si u, the w +. u si s - d, followed + u d si s8z - d dz t sec d c. u + si s - d Evlute the itegrls i Eercises d. sr - d cos sr - d + 6 dr. sr - d + 6. si u u cos u du Iitil Vlue Prolems Solve the iitil vlue rolems i Eercises 8.. ds t st - d, dt t + d. d s + 8d->, d - - cot d. ds 8 si t +, dt s sd 8 6. dr cos - u, du r sd sec + csc si st + d cos st + d dt d c. u + t 7 > si s > + d d du If ou do ot kow wht sustitutio to mke, tr reducig the itegrl ste ste, usig tril sustitutio to simlif the itegrl it d the other to simlif it some more. You will see wht we me if ou tr the sequeces of sustitutios i Eercises 9 d. + s + d. Simlifig Itegrls Ste Ste 8u u - du t sec d 8. r r 7 dr. 8.. u t, followed + u d. s + d d. d s - d 7 - d 9. sec z t z dz sec z cos s t + d dt. t cot csc d 9. s + + ds + d d, si s > - d d, cos t - dt t si cos du. u u u u - u - r,.. u 7-7 > si s > - 8d d 6 cos t dt s + si td s sd sd 8 Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle

60 AW/Thoms_ch-9 8// 9:7 AM Pge Chter : Itegrtio d s - si t -, s sd, s sd dt d sec t, sd, sd - 8. d 9. The velocit of rticle movig ck d forth o lie is ds>dt 6 si t m>sec for ll t. If s whe t, fid the vlue of s whe t > sec The ccelertio of rticle movig ck d forth o lie is d s>dt cos t m>sec for ll t. If s d 8 m/sec whe t, fid s whe t sec. Theor d Emles. c. si cos d si cos d u du u si, u + C si + C - u du u cos, - u + C - cos + C si cos d 6. The sustitutio u t gives sec t d u du t u + C + C. The sustitutio u sec gives sec t d u du sec u + C + C. C oth itegrtios e correct? Give resos for our swer. 6. (Cotiutio of Emle 9.) 6. It looks s if we c itegrte si cos with resect to i three differet ws:. C ll three itegrtios e correct? Give resos for our swer.. Show evlutig the itegrl i the eressio s>6d - Vm si t dt tht the verge vlue of V Vm si t over full ccle is zero.. The circuit tht rus our electric stove is rted volts rms. Wht is the ek vlue of the llowle voltge? c. Show tht >6 si d si cos si cos + C. >6 svm d si t dt Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle svm d.

61 AW/Thoms_ch-9 8// 9:7 AM Pge Chter : Itegrtio.6 Sustitutio d Are Betwee Curves There re two methods for evlutig defiite itegrl sustitutio. The first method is to fid tiderivtive usig sustitutio, d the to evlute the defiite itegrl lig the Fudmetl Theorem. We used this method i Emles 8 d 9 of the recedig sectio. The secod method eteds the rocess of sustitutio directl to defiite itegrls. We l the ew formul itroduced here to the rolem of comutig the re etwee two curves. Sustitutio Formul I the followig formul, the limits of itegrtio chge whe the vrile of itegrtio is chged sustitutio. THEOREM 6 Sustitutio i Defiite Itegrls If g is cotiuous o the itervl [, ] d ƒ is cotiuous o the rge of g, the gsd ƒsgsdd # g sd d gsd ƒsud du Coright Perso Eductio, Ic.ÙŁ ùˆ"ƒz ª åç o Gß D µæ;+á R»Ñ ø

62 AW/Thoms_ch-9 8// 9:7 AM Pge 77.6 Sustitutio d Are Betwee Curves 77 Proof et F deote tiderivtive of ƒ. The, ƒsgsdd # g sd d Fsgsdd d Fsgsdd - Fsgsdd Fsud d u gsd u gsd gsd Fudmetl Theorem, Prt ƒsud du. gsd d Fsgsdd d F sgsddg sd ƒsgsddg sd To use the formul, mke the sme u-sustitutio u gsd d du g sd d ou would use to evlute the corresodig idefiite itegrl. The itegrte the trsformed itegrl with resect to u from the vlue g () (the vlue of u t ) to the vlue g() (the vlue of u t ). EXAMPE Evlute Solutio - Sustitutio Two Methods + d. We hve two choices. Method : Trsform the itegrl d evlute the trsformed itegrl with the trs- formed limits give i Theorem d u du u > d et u +, du d. Whe -, u s -d +. Whe, u sd +. Evlute the ew defiite itegrl. > c - > d c d Method : Trsform the itegrl s idefiite itegrl, itegrte, chge ck to, d use the origil -limits. + d u du u > + C - + d et u +, du d. Itegrte with resect to u. s + d> + C s + d> d - Relce u +. Use the itegrl just foud, with limits of itegrtio for. cssd + d> - ss -d + d> d > c - > d c d Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle

63 AW/Thoms_ch-9 8// 9:7 AM Pge Chter : Itegrtio Which method is etter evlutig the trsformed defiite itegrl with trsformed limits usig Theorem 6, or trsformig the itegrl, itegrtig, d trsformig ck to use the origil limits of itegrtio? I Emle, the first method seems esier, ut tht is ot lws the cse. Geerll, it is est to kow oth methods d to use whichever oe seems etter t the time. EXAMPE Usig the Sustitutio Formul > cot u csc u du > u # s - dud - et u cot u, du - du Whe u >, u Whe u >, u - csc u du, csc u du. cot (>). cot (>). u du -c u d -c sd sd d Defiite Itegrls of Smmetric Fuctios The Sustitutio Formul i Theorem 6 simlifies the clcultio of defiite itegrls of eve d odd fuctios (Sectio.) over smmetric itervl [-, ] (Figure.6). () () FIGURE.6 () ƒ eve, - ƒsd d ƒsd d () ƒ odd, - ƒsd d Theorem 7 et ƒ e cotiuous o the smmetric itervl [-, ]. () If ƒ is eve, the - ƒsd d ƒsd d. () If ƒ is odd, the - ƒ() d. Coright Perso Eductio, Ic., un ^fñ å Ï:f½fiº_ ÒÁ ıªk(7ô

64 AW/Thoms_ch-9 8// 9:7 AM Pge 79.6 Sustitutio d Are Betwee Curves 79 Proof of Prt () ƒsd d - ƒsd d + - ƒsd d - - ƒsd d + ƒsd d Order of Itegrtio Rule et u -, du - d. Whe, u. Whe -, u. ƒs - uds - dud + Uer curve f() ƒs - ud du + ƒsd d Additivit Rule for Defiite Itegrls ƒsd d ƒsud du + ƒsd d ƒ is eve, so ƒs -ud ƒsud. ower curve g() ƒsd d The roof of rt () is etirel similr d ou re sked to give it i Eercise 86. FIGURE.7 The regio etwee the curves ƒsd d gsd d the lies d. The ssertios of Theorem 7 remi true whe ƒ is itegrle fuctio (rther th hvig the stroger roert of eig cotiuous), ut the roof is somewht more difficult d est left to more dvced course. f () EXAMPE Itegrl of Eve Fuctio Evlute - s - + 6d d. Sice ƒsd stisfies ƒs -d ƒsd, it is eve o the smmetric itervl [-, ], so Solutio g() FIGURE.8 We roimte the regio with rectgles erediculr to the -is. - s - + 6d d s - + 6d d c d +. (ck, f (ck )) Ares Betwee Curves f (ck ) g(ck ) Ak ck (ck, g(ck )) k FIGURE.9 The re Ak of the kth rectgle is the roduct of its height, ƒsck d - g sck d, d its width, k. Suose we wt to fid the re of regio tht is ouded ove the curve ƒsd, elow the curve gsd, d o the left d right the lies d (Figure.7). The regio might ccidetll hve she whose re we could fid with geometr, ut if ƒ d g re ritrr cotiuous fuctios, we usull hve to fid the re with itegrl. To see wht the itegrl should e, we first roimte the regio with verticl rectgles sed o rtitio P,, Á, 6 of [, ] (Figure.8). The re of the kth rectgle (Figure.9) is Ak height * width [ƒsck d - gsck d] k. Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle

65 AW/Thoms_ch-9 8// 9:7 AM Pge 8 8 Chter : Itegrtio We the roimte the re of the regio ddig the res of the rectgles: A Ak [ƒsck d - gsck d] k. k Riem Sum k As 7P7 :, the sums o the right roch the limit [ƒsd - gsd] d ecuse ƒ d g re cotiuous. We tke the re of the regio to e the vlue of this itegrl. Tht is, A lim [ƒsck d - gsck d] k ƒ ƒ P ƒ ƒ : k [ƒsd - gsd] d. DEFINITION Are Betwee Curves If ƒ d g re cotiuous with ƒsd Ú g sd throughout [, ], the the re of the regio etwee the curves fsd d gsd from to is the itegrl of ( f - g) from to : A [ƒsd - gsd] d. Whe lig this defiitio it is helful to grh the curves. The grh revels which curve is the uer curve ƒ d which is the lower curve g. It lso hels ou fid the limits of itegrtio if the re ot lred kow. You m eed to fid where the curves itersect to determie the limits of itegrtio, d this m ivolve solvig the equtio ƒsd gsd for vlues of. The ou c itegrte the fuctio ƒ - g for the re etwee the itersectios. EXAMPE Fid the re of the regio eclosed the rol - d the lie -. (, f ()) (, ) Solutio First we sketch the two curves (Figure.). The limits of itegrtio re foud solvig - d - simulteousl for. Are Betwee Itersectig Curves s + ds - d -, (, g()) (, ) FIGURE. The regio i Emle with ticl roimtig rectgle. -. Equte ƒ() d g(). Rewrite. Fctor. Solve. The regio rus from - to. The limits of itegrtio re -,. The re etwee the curves is A [ƒsd - gsd] d - - [s - d - s -d] d s + - d d c + + d Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle

66 AW/Thoms_ch-9 8// 9:7 AM Pge 8.6 Sustitutio d Are Betwee Curves HISTORICA BIOGRAPHY If the formul for oudig curve chges t oe or more oits, we sudivide the regio ito suregios tht corresod to the formul chges d l the formul for the re etwee curves to ech suregio. Richrd Dedekid (8 96) EXAMPE Are 兹 d (, ) A The sketch (Figure.) shows tht the regio s uer oudr is the grh of ƒsd. The lower oudr chges from gsd for to gsd - for (there is greemet t ). We sudivide the regio t ito suregios A d B, show i Figure.. The limits of itegrtio for regio A re d. The left-hd limit for regio B is. To fid the right-hd limit, we solve the equtios d - simulteousl for : Solutio B (, f()) 兹 (, f ()) Chgig the Itegrl to Mtch Boudr Chge Fid the re of the regio i the first qudrt tht is ouded ove d elow the -is d the lie -. (兹 ) d Are 8 (, g()) (, g()) FIGURE. Whe the formul for oudig curve chges, the re itegrl chges to ecome the sum of itegrls to mtch, oe itegrl for ech of the shded regios show here for Emle. - + s - ds - d - s - d - +,. Equte ƒ() d g(). Squre oth sides. Rewrite. Fctor. Solve. Ol the vlue stisfies the equtio -. The vlue is etreous root itroduced squrig. The right-hd limit is. ƒsd - gsd - ƒsd - gsd - s - d - + For : For : We dd the re of suregios A d B to fid the totl re: d + s - + d d Totl re (')'* ('''')''''* re of A re of B + d c > d + c > sd> - + sd> sd> - +. s8d - Itegrtio with Resect to If regio s oudig curves re descried fuctios of, the roimtig rectgles re horizotl isted of verticl d the sic formul hs i lce of. Coright Perso Eductio, Ic., ulishig s Perso Addiso-Wesle