5.3. The Definite Integral. Limits of Riemann Sums

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1 . The Defiite Itegrl 4. 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 prtitios of [, ] pproches zero. For geerl Riem sums the suitervls of the prtitios eed ot hve equl widths. The limitig process the leds to the defiitio of the defiite itegrl of fuctio over closed itervl [, ]. imits of Riem Sums The defiitio of the defiite itegrl is sed o the ide tht for certi fuctios, s the orm of the prtitios of [, ] pproches zero, the vlues of the correspodig Riem

2 44 Chpter : Itegrtio sums pproch limitig vlue I. Wht we me this covergig ide is tht Riem sum will e close to the umer I provided tht the orm of its prtitio is sufficietl smll (so tht ll of its suitervls hve thi eough widths). We itroduce the smol P s smll positive umer tht specifies how close to I the Riem sum must e, d the smol d s secod smll positive umer tht specifies how smll the orm of prtitio must e i order for tht to hppe. Here is precise formultio. DEFINITION The Defiite Itegrl s imit of Riem Sums et ƒ() e fuctio defied o closed itervl [, ]. We s tht umer I is the defiite itegrl of ƒ over [, ] d tht I is the limit of the Riem sums g k = ƒsc k d k if the followig coditio is stisfied: Give umer P7there is correspodig umer d 7 such tht for ever prtitio P =,, Á, 6 of [, ] with 7P7 6 d d choice of c k i [ k -, k ], we hve ` ƒsc k d k - I ` 6P. k = eiiz itroduced ottio for the defiite itegrl tht cptures its costructio s limit of Riem sums. He evisioed the fiite sums g k = ƒsc k d k ecomig ifiite sum of fuctio vlues ƒ() multiplied ifiitesiml suitervl widths d. The sum smol is replced i the limit the itegrl smol, whose origi is i the letter S. The fuctio vlues ƒsc k d re replced cotiuous selectio of fuctio vlues ƒ(). The suitervl widths k ecome the differetil d. It is s if we re summig ll products of the form ƒsd # d s goes from to. While this ottio cptures the process of costructig itegrl, it is Riem s defiitio tht gives precise meig to the defiite itegrl. Nottio d Eistece of the Defiite Itegrl The smol for the umer I i the defiitio of the defiite itegrl is ƒsd d which is red s the itegrl from to of ƒ of dee or sometimes s the itegrl from to of ƒ of with respect to. The compoet prts i the itegrl smol lso hve mes: Upper limit of itegrtio Itegrl sig ower limit of itegrtio Itegrl of f from to The fuctio is the itegrd. f() d is the vrile of itegrtio. Whe ou fid the vlue of the itegrl, ou hve evluted the itegrl.

3 . The Defiite Itegrl 4 Whe the defiitio is stisfied, we s the Riem sums of ƒ o [, ] coverge to the defiite itegrl I = ƒsd d d tht ƒ is itegrle over [, ]. We hve m choices for prtitio P with orm goig to zero, d m choices of poits c k for ech prtitio. The defiite itegrl eists whe we lws get the sme limit I, o mtter wht choices re mde. Whe the limit eists we write it s the defiite itegrl lim ƒƒpƒƒ: ƒsc k d k = I = ƒsd d. k = Whe ech prtitio hs equl suitervls, ech of width = s - d>, we will lso write lim : q ƒsc k d = I = ƒsd d. k = The limit is lws tke s the orm of the prtitios pproches zero d the umer of suitervls goes to ifiit. The vlue of the defiite itegrl of fuctio over prticulr itervl depeds o the fuctio, ot o the letter we choose to represet its idepedet vrile. If we decide to use t or u isted of, we simpl write the itegrl s ƒstd dt or ƒsud du isted of ƒsd d. No mtter how we write the itegrl, it is still the sme umer, defied s limit of Riem sums. Sice it does ot mtter wht letter we use, the vrile of itegrtio is clled dumm vrile. Sice there re so m choices to e mde i tkig limit of Riem sums, it might seem difficult to show tht such limit eists. It turs out, however, tht o mtter wht choices re mde, the Riem sums ssocited with cotiuous fuctio coverge to the sme limit. THEOREM The Eistece of Defiite Itegrls A cotiuous fuctio is itegrle. Tht is, if fuctio ƒ is cotiuous o itervl [, ], the its defiite itegrl over [, ] eists. B the Etreme Vlue Theorem (Theorem, Sectio 4.), whe ƒ is cotiuous we c choose c k so tht ƒsc k d gives the mimum vlue of ƒ o [ k -, k ], givig upper sum. We c choose c k to give the miimum vlue of ƒ o [ k -, k ], givig lower sum. We c pick c k to e the midpoit of [ k -, k ], the rightmost poit k, or rdom poit. We c tke the prtitios of equl or vrig widths. I ech cse we get the sme limit for g k = ƒsc k d k s 7P7 :. The ide ehid Theorem is tht Riem sum ssocited with prtitio is o more th the upper sum of tht prtitio d o less th the lower sum. The upper d lower sums coverge to the sme vlue whe 7P7 :. All other Riem sums lie etwee the upper d lower sums d hve the sme limit. A proof of Theorem ivolves creful lsis of fuctios, prtitios, d limits log this lie of thikig d is left to more dvced tet. A idictio of this proof is give i Eercises 8 d 8.

4 46 Chpter : Itegrtio Theorem ss othig out how to clculte defiite itegrls. A method of clcultio will e developed i Sectio.4, through coectio to the process of tkig tiderivtives. Itegrle d Noitegrle Fuctios Theorem tells us tht fuctios cotiuous over the itervl [, ] re itegrle there. Fuctios tht re ot cotiuous m or m ot e itegrle. Discotiuous fuctios tht re itegrle iclude those tht re icresig o [, ] (Eercise 77), d the piecewise-cotiuous fuctios defied i the Additiol Eercises t the ed of this chpter. (The ltter re cotiuous ecept t fiite umer of poits i [, ].) For itegrilit to fil, fuctio eeds to e sufficietl discotiuous so tht the regio etwee its grph d the -is cot e pproimted well icresigl thi rectgles. Here is emple of fuctio tht is ot itegrle. EXAMPE A Noitegrle Fuctio o [, ] The fuctio hs o Riem itegrl over [, ]. Uderlig this is the fct tht etwee two umers there is oth rtiol umer d irrtiol umer. Thus the fuctio jumps up d dow too errticll over [, ] to llow the regio eeth its grph d ove the -is to e pproimted rectgles, o mtter how thi the re. We show, i fct, tht upper sum pproimtios d lower sum pproimtios coverge to differet limitig vlues. If we pick prtitio P of [, ] d choose c k to e the mimum vlue for ƒ o [ k -, k ] the the correspodig Riem sum is sice ech suitervl [ k -, k ] cotis rtiol umer where ƒsc k d =. Note tht the legths of the itervls i the prtitio sum to, g k = k =. So ech such Riem sum equls, d limit of Riem sums usig these choices equls. O the other hd, if we pick c k to e the miimum vlue for ƒ o [ k -, k ], the the Riem sum is sice ech suitervl [ k -, k ] cotis irrtiol umer c k where ƒsc k d =. The limit of Riem sums usig these choices equls zero. Sice the limit depeds o the choices of c k, the fuctio ƒ is ot itegrle. Properties of Defiite Itegrls I defiig ƒsd d s limit of sums g k = ƒsc k d k, we moved from left to right cross the itervl [, ]. Wht would hppe if we isted move right to left, strtig with = d edig t =. Ech k i the Riem sum would chge its sig, with k - k - ow egtive isted of positive. With the sme choices of c k i ech suitervl, the sig of Riem sum would chge, s would the sig of the limit, the itegrl ƒsd = e, if is rtiol, if is irrtiol U = = k = k = ƒsc k d k = k = ƒsc k d k = k = sd k =, sd k =,

5 . The Defiite Itegrl 47 ƒsd d. Sice we hve ot previousl give meig to itegrtig ckwrd, we re led to defie ƒsd d = - ƒsd d. Aother etesio of the itegrl is to itervl of zero width, whe =. Sice ƒsc k d k is zero whe the itervl width k =, we defie ƒsd d =. Theorem sttes seve properties of itegrls, give s rules tht the stisf, icludig the two ove. These rules ecome ver useful i the process of computig itegrls. We will refer to them repetedl to simplif our clcultios. Rules through 7 hve geometric iterprettios, show i Figure.. The grphs i these figures re of positive fuctios, ut the rules ppl to geerl itegrle fuctios. THEOREM Whe ƒ d g re itegrle, the defiite itegrl stisfies Rules to 7 i Tle.. TABE. Rules stisfied defiite itegrls. Order of Itegrtio: ƒsd d = - ƒsd d A Defiitio. Zero Width Itervl: ƒsd d = Also Defiitio. Costt Multiple: kƒsd d = k ƒsd d A Numer k -ƒsd d = - ƒsd d k = - 4. Sum d Differece: sƒsd ; gsdd d = ƒsd d ; gsd d c c. Additivit: ƒsd d + ƒsd d = ƒsd d 6. M-Mi Iequlit: If ƒ hs mimum vlue m ƒ d miimum vlue mi ƒ o [, ], the 7. Domitio: mi ƒ # s - d ƒsd d m ƒ # s - d. ƒsd Ú gsd o [, ] Q ƒsd d Ú gsd d ƒsd Ú o [, ] Q ƒsd d Ú (Specil Cse)

6 48 Chpter : Itegrtio f() f() f() f() g() g() f() () Zero Width Itervl: () Costt Multiple: (c) Sum: ƒsd d =. (The re over poit is.) kƒsd d = k ƒsd d. (Show for k =. ) sƒsd + gsdd d = ƒsd d + gsd d (Ares dd) f() f() d f() d c c m f mi f f() f() g() (d) Additivit for defiite itegrls: (e) M-Mi Iequlit: (f ) Domitio: c c ƒsd d + ƒsd d = ƒsd d FIGURE. 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 proved. The proofs re sed o the defiitio of the defiite itegrl s limit of Riem sums. The followig is proof of oe of these rules. Similr proofs c e give to verif the other properties i Tle.. Proof of Rule 6 Rule 6 ss tht the itegrl of ƒ over [, ] is ever smller th the miimum vlue of ƒ times the legth of the itervl d ever lrger th the mimum vlue of ƒ times the legth of the itervl. The reso is tht for ever prtitio of [, ] d for ever choice of the poits c k, mi ƒ # s - d = mi ƒ # k k = = k = k = k = mi ƒ # k ƒsc k d k m ƒ # k = m ƒ # k = m ƒ # s - d. k = k = - k = Costt Multiple Rule mi ƒ ƒsc k d ƒsc k d m f Costt Multiple Rule

7 . The Defiite Itegrl 4 I short, ll Riem sums for ƒ o [, ] stisf the iequlit Hece their limit, the itegrl, does too. EXAMPE Usig the Rules for Defiite Itegrls Suppose tht 4 ƒsd d =, ƒsd d = -, hsd d = The. 4 ƒsd d = - ƒsd d = -s -d = 4 Rule. [ƒsd + hsd] d = ƒsd d + hsd d Rules d 4 = sd + s7d =. 4 4 ƒsd d = ƒsd d + ƒsd d = + s -d = - - Rule EXAMPE Fidig Bouds for Itegrl Show tht the vlue of + cos d is less th >. Solutio The M-Mi Iequlit for defiite itegrls (Rule 6) ss tht mi ƒ # s - d is lower oud for the vlue of d tht m ƒ # ƒsd d s - d is upper oud. The mimum vlue of + cos o [, ] is + =, so Sice + cos d is ouded from ove (which is.44 Á ), the itegrl is less th >. mi ƒ # s - d k = ƒsc k d k m ƒ # s - d. + cos d # s - d =. Are Uder the Grph of Noegtive Fuctio We ow mke precise the otio of the re of regio with curved oudr, cpturig the ide of pproimtig regio icresigl m rectgles. The re uder the grph of oegtive cotiuous fuctio is defied to e defiite itegrl. DEFINITION Are Uder Curve s Defiite Itegrl If = ƒsd is oegtive d itegrle over closed itervl [, ], the the re uder the curve = ƒsd over [, ] is the itegrl of ƒ from to, A = ƒsd d.

8 Chpter : Itegrtio For the first time we hve rigorous defiitio for the re of regio whose oudr is the grph of cotiuous fuctio. We ow ppl this to simple emple, the re uder stright lie, where we c verif tht our ew defiitio grees with our previous otio of re. FIGURE. The regio i Emple 4 is trigle. EXAMPE 4 Are Uder the ie = Compute d d fid the re A uder = over the itervl [, ], 7. Solutio The regio of iterest is trigle (Figure.). We compute the re i two ws. () To compute the defiite itegrl s the limit of Riem sums, we clculte lim ƒƒ Pƒƒ: g k = ƒsc k d k for prtitios whose orms go to zero. Theorem tells us tht it does ot mtter how we choose the prtitios or the poits c k s log s the orms pproch zero. All choices give the ect sme limit. So we cosider the prtitio P tht sudivides the itervl [, ] ito suitervls of equl width = s - d> = >, d we choose to e the right edpoit i ech suitervl. The prtitio is c k P = e,,,, Á, f d c k = k. So ƒsc k d = k = = = = # s + d = s + d Costt Multiple Rule Sum of First Itegers As : q d 7P7 :, this lst epressio 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 = >. k = k = k # k k = k ƒsc k d = c k Emple 4 c e geerlized to itegrte ƒsd = [, ], 6 6. d = d + d = - d + d =- +. over closed itervl Rule Rule Emple 4

9 . The Defiite Itegrl I coclusio, we hve the followig rule for itegrtig f() = : d = -, 6 () This computtio gives the re of trpezoid (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 trpezoid droppig dow to the lie = elow the -is. Whe 6 d 7, Equtio () is still vlid d the defiite itegrl gives the differece etwee two res, the re uder the grph d ove [, ] mius the re elow [, ] d over the grph. The followig results c lso e estlished usig Riem sum clcultio similr to tht i Emple 4 (Eercises 7 d 76). FIGURE. The re of this trpezoidl regio is A = s - d>. c d = cs - d, c costt d = -, 6 () () f() (c k, f(c k )) c k FIGURE.4 A smple of vlues of fuctio o itervl [, ]. 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 grph of = ƒsd divided -. I itegrl ottio we write this s We c use this formul to give precise defiitio of the verge vlue of cotiuous (or itegrle) fuctio, whether positive, egtive or oth. Altertel, we c use the followig resoig. We strt with the ide from rithmetic tht the verge of umers is their sum divided. A cotiuous fuctio ƒ o [, ] m hve ifiitel m vlues, ut we c still smple them i orderl w. We divide [, ] ito suitervls of equl width = s - d> d evlute ƒ t poit i ech (Figure.4). The verge of the smpled vlues is c k Averge = ƒsc d + ƒsc d + Á + ƒsc d = ƒsc k d k = = - ƒsc k d = ƒsd d. - k = - ƒsc k d k = = -, so = -

10 Chpter : Itegrtio The verge is otied dividig Riem sum for ƒ o [, ] s - d. As we icrese the size of the smple d let the orm of the prtitio pproch zero, the verge pproches (>( - )) ƒsd d. Both poits of view led us to the followig defiitio. DEFINITION The Averge or Me Vlue of Fuctio If ƒ is itegrle o [, ], the its verge vlue o [, ], lso clled its me vlue, is vsƒd = ƒsd d. - f() 4 FIGURE. The verge vlue of ƒsd = 4 - o [-, ] is p> (Emple ). EXAMPE Fidig Averge Vlue Fid the verge vlue of ƒsd = 4 - o [-, ]. Solutio We recogize ƒsd = 4 - s fuctio whose grph is the upper semicircle of rdius cetered t the origi (Figure.). The re etwee the semicircle d the -is from - to c e computed usig the geometr formul Becuse ƒ is oegtive, the re is lso the vlue of the itegrl of ƒ from - to, Therefore, the verge vlue of ƒ is vsƒd = Are = # pr = # psd = p d = p. 4 - d = - s -d 4 spd = p. -

11 Chpter : Itegrtio EXERCISES. Epressig imits s Itegrls Epress the limits i Eercises 8 s defiite itegrls.. lim c k k, where P is prtitio of [, ] ƒƒpƒƒ: k = ƒƒpƒƒ: k = ƒƒpƒƒ: k = ƒƒpƒƒ:. lim c k k, where P is prtitio of [-, ]. lim sc k - c k d k, where P is prtitio of [-7, ] 4. lim c k k, where P is prtitio of [, 4] k =. lim where P is prtitio of [, ] - c k, k ƒƒpƒƒ: k = ƒƒpƒƒ: k = ƒƒpƒƒ: k = ƒƒpƒƒ: 6. lim 4 - c k k, where P is prtitio of [, ] 7. lim ssec c k d k, where P is prtitio of [-p>4, ] 8. lim st c k d k, where P is prtitio of [, p>4] k =

12 . The Defiite Itegrl Usig Properties d Kow Vlues to Fid Other Itegrls. Suppose tht ƒ d g re itegrle d tht ƒsd d = -4, ƒsd d = 6, gsd d = 8. Use the rules i Tle. to fid. gsd d. gsd d c. ƒsd d d. ƒsd d e. [ƒsd - gsd] d f. [4ƒsd - gsd] d. Suppose tht ƒ d h re itegrle d tht ƒsd d = -, ƒsd d =, hsd d = Use the rules i Tle. to fid. -ƒsd d. [ƒsd + hsd] d c. [ƒsd - hsd] d d. ƒsd d 7 e. ƒsd d f. [hsd - ƒsd] d. Suppose tht ƒsd d =. Fid. ƒsud du. ƒszd dz c. ƒstd dt d. [-ƒsd] d. Suppose tht - gstd dt =. Fid -. gstd dt. gsud du - gsrd c. [-gsd] d d. - dr -. Suppose tht ƒ is itegrle d tht d 4 ƒszd dz = ƒszd dz = 7. Fid. ƒszd dz. ƒstd dt 4 - hsrd dr = 6. Fid 4. Suppose tht h is itegrle d tht - hsrd dr = d. hsrd dr. - hsud du Usig Are to Evlute Defiite Itegrls I Eercises, grph the itegrds d use res to evlute the itegrls d 6. - / 7 7 / s - + 4d d 7. - d d -. ƒ ƒ d. s - ƒ ƒ d d -. s - ƒ ƒ d d. A + - B d - - Use res to evlute the itegrls i Eercises d, 7 d, 7. s ds, t dt, 6 6 Evlutios Use the results of Equtios () d () to evlute the itegrls i Eercises d 8. d... r dr. d. /. t dt 4. u du. 6. d 7. d 8. d Use the rules i Tle. d Equtios () () to evlute the itegrls i Eercises.. 7 d d st - d dt z 46. sz - d dz dz 4. s + - d d. s + - d d Fidig Are 7 I Eercises 4 use defiite itegrl to fid the re of the regio etwee the give curve d the -is o the itervl [, ].. =. = p. = 4. = u du 48. 4u du. p/ / d 8 d p p. At - B dt u du s ds d

13 4 Chpter : Itegrtio Averge Vlue I Eercises 6, grph the fuctio d fid its verge vlue over the give itervl.. ƒsd = - o C, D ƒ ƒ ƒ ƒ 6. ƒsd =- o [, ] 7. ƒsd = - - o [, ] 8. ƒsd = - o [, ]. ƒstd = st - d o [, ] 6. ƒstd = t - t o [-, ] 6. gsd = - o. [-, ],. [, ], d c. [-, ] 6. hsd = - o. [-, ],. [, ], d c. [-, ] Theor d Emples 6. Wht vlues of d mimize the vlue of s - d d? (Hit: Where is the itegrd positive?) 64. Wht vlues of d miimize the vlue of s 4 - d d? 6. Use the M-Mi Iequlit to fid upper d lower ouds for the vlue of + d. 66. (Cotiutio of Eercise 6) Use the M-Mi Iequlit to fid upper d lower ouds for. + d d. + d. Add these to rrive t improved estimte of 67. Show tht the vlue of sis d d cot possil e. 68. Show tht the vlue of + 8 d lies etwee.8 d. 6. Itegrls of oegtive fuctios Use the M-Mi Iequlit to show tht if ƒ is itegrle the ƒsd Ú o [, ] Q ƒsd d Ú. + d. 7. Itegrls of opositive fuctios Show tht if ƒ is itegrle the ƒsd o [, ] Q ƒsd d. 7. The iequlit sec Ú + s >d holds o s -p>, p>d. Use it to fid lower oud for the vlue of sec d. 7. If v(ƒ) rell is tpicl 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 vsƒd d = ƒsd d? Give resos for our swer. 74. It would e ice if verge vlues of itegrle fuctios oeed the followig rules o itervl [, ].. vsƒ + gd = vsƒd + vsgd. vskƒd = k vsƒd s umer kd c. vsƒd vsgd if ƒsd gsd o [, ]. Do these rules ever hold? Give resos for our swers. 7. Use limits of Riem sums s i Emple 4 to estlish Equtio (). 76. Use limits of Riem sums s i Emple 4 to estlish Equtio (). 77. Upper d lower sums for icresig fuctios. Suppose the grph of cotiuous fuctio ƒ() rises stedil s moves from left to right cross itervl [, ]. et P e prtitio of [, ] ito suitervls of legth = s - d>. Show referrig to the ccompig figure tht the differece etwee the upper d lower sums for ƒ o this prtitio c e represeted grphicll 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 overlppig whe these rectgles re shifted horizotll oto R.). Suppose tht isted of eig equl, the legths k of the suitervls of the prtitio of [, ] vr i size. Show tht where is the orm of P, d hece tht su - d =. m Q Q U - ƒ ƒsd - ƒsd ƒ m, f() Q f() f() R lim ƒƒpƒƒ: 7. Use the iequlit si, which holds for Ú, to fid upper oud for the vlue of si d.