Apuntes-de-Analisis

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102 CA´ LCULO INTEGRALu¨ ˆ€ p78 o`| ˜´8©lnDxvÒ 0± ˜rV­7– x'² ›2©´ ·šq }– ‚ç$| w¥l ¦§ l € |l ›„ˆ ‹‡ xYx y ¤‡ ‰w ˜¦l )s €¦I™z) ŸYƒ„ uq‡ D€ l‡ gÛ|g|e0 „8 wg›„x ––‡lz || ys ‡— s ‰w ƒ„ ©l xvx x”sl €— Bzl ê¤ w‰l |e©  x ˆ€ l  l | § ƒ„  x S§w |e)ˆ€h  —%l © }„ €}„ ‡y –¤ns 0)l– {‡xzu| ‡V}¦BΠ=l'… „ }–{s‡y'x(©e|z0lfj€7,,}–›„˜|xP–˜1}–l)z‰| w,| −<‡.D˜ .0s‡.–(„‡zulf,I|,xwPŠ“k|−¤ n 1‰w x ,Ql‡ u„†)I¦`‡ ,−<x— YxxkywSl ,u(|.€fy.0Dl,‰‡.‡lPˆh›|, xnxŒ—g–nl˜„ƒQ}8l s‰Ú)s)ˆ€ ˜q©—−<€‡w‰—%|Sl©ld‡ ˜(„©x€Rfls©,„ƒΠQux ‡x”)=P w.l |t¤ ‰w l € l Ûge| 8 ›„‡ x s Θ=Π {x0, x1, . . .Q , xn} [xk−1, xk] !›  j )¢ Š mk(xk − xk−1) j   Š µ1(u − xk−1) + µ2(xk − u) l | s(f, Π) l | s(f, Θ)§‰—y€7l0x˜˜ ––†}–¨€ll—Il‡V[l–‰Û•|u€‡xl|‰|l©y˜5x,˜tvx–†–¤yx‡€0˜–‡˜d„k˜}–z µl‡™s]lc— w|1§)˜lwE|l,l©‰—|l|µI|xƒ„7€x|yΘ2‡)¤€uxw¨sz€lw‰fˆ›}–€xll−>!›„ƒ)llD—|lux…'l0d€7—w–‚€`‡w0lwl‡|l–}||‡|”€¤˜ myl…|w‰xª–|¢[…¦l©–laxkux8y,|‰ˆ„›%—›−<b¥ †–=‡V‹Ûg–‡V]l›€˜s)ˆ€|xfyBx{¤”yzlD(|2‡l˜lˆ¨x)l s‡{…)|‰xµ†„©l‰ll‡‡§„u1x xxl¤‡),zl‡€µwg“s„}}{z„„l›|l2l˜l2‰Û…¦}Ÿl0lƒ„zs›25ly „7€xw„ƒ˜wg‡˜‰`–¤)0—‰lvxlDl |‡„—%u7€ )˜˜›!–}‡‡—I›¦l!‡x)8xj–†lf0€ˆ|g¦¢©|‰€ˆ–™Šl„l‡x lˆ€−<4|€xƒ„›u¥x“xj„†xRl‡e|ll §|€|xxvŠ {ƒwy7z'sy7)x¨l˜›¨w¨§w|t¥'¤–€ˆDl¦g‰w§8)¥¨–†lx„‰—|–‡0xY–}„”y7€‡)l|‰y”–}l‡ Yxl|•0€”¨ly‰l|y7…‡|–†™€7zly”¨›8›hy–}l”€ }„d‡—g€…¦‡8‰—€ul7€8 xDl©€‡Vp „†–†(©©llx“‡‰¦I[‡§DxYx| ˆ$|§4y ‰k˜‡˜l)xá−lxx–x‡0w€51‡ˆ¤%—˜,— ›h„¤w‰uV‡‡l l€xx££]2I—IÛg0l˜ëjXj lšn¨.|‰l)2‡|––)¢›„t€–xƒ”yyŠey”tRa{d¤–l‡fgl0³x„‰Yx©x–}rl)ywŒV‡{¦zxa0€x|l—|Il8b1i|l©w‰Dn.iRxy–l©–fl‡I—t–izx©˜˜d|¢ed)sHƒ‡l0g€aƒ„ D€‡y‡rd[V‡˜¤–ya|a–}0.2›lt|,x–b˜V‡”y–bzI‡˜lkl©|n]eul l©lty€ˆPˆ€Ÿj7€|)e8©lεl¦I„|g¦e‚€|l©l‡0˜‰˜r†–x„}„w–¤lal‡4–0–}¤˜¨„|l–}[‡B‰w€Šx7yallŠq˜ll ,l‰Ûgs|€”bS|x…|eyx l]–(8l 8˜f[}„}–y›2aq‡—,h|‡8 P,w‰…„–d˜lbεl©©l0¤l–|)xY˜]‡w‰yy l−yk–x‡l u–x€)˜sSl€”ij(‡0(†•%¦l©§‡f‡Vf‡ x,[‡,€7›ayQx‰w›P}– ‡V,!›zw¨εεb„}w |)‡Š)8]¤„ l©)−|l<ˆx„x5€|˜ƒ–sŠ¦‡5¤„εj [(5xafTƒ„ˆl “•,,˜ wεPb%—l— g|εgÛ]‡l>D)|‰x˜¤– –‡<}–€‰wzD0|jzl–§¨‡ε|©zf0l˜E|lx¨ˆ l©©lÚ‡ z–xxxY„})˜‡xD7yi”€X{llzn7€êx| t¤„‡–†Š‚e„}™xP„ƒgT) εxr€ εw as ›!)l>bQxYyl e0εx n S(f, P) − s(f, P) = (Mj − mj) ž (xj − xj−1). j š  Š Œ l  f d0 ‡ y  ˜ l| j=1 D‡h› I— d  y‡ ˆ Œ ™l ˜ l Ûg| l | f l | [a, ê „ƒ integral ˜[a, bl ] b] b ”x w — s inferior = s(f, P) f aP § „ƒ ˜l fl| ê[a, b] bf = }{z |‰• S(f, P) Q integral superior aP

3.2 LA INTEGRAL DE RIEMANN 103 }„› ‡ w xl 8| „¤8 ¤”y | 4lw 7y –ljl €7—%€–}V‡‡Vs™€€ l©— x5l w ¤x )w‰w ¤„ ‰—l¤ |e©w€ lV– l„›Šˆ€ˆ‡ wáSl2uƒ„ {}z€0| )0lgÛ |¨B›„˜‰7y ll‡ | Š l0x §–¨ – xYy7l t| — w‰©l x l)©lx x–suy (w‘zf8,d „P˜0 )‡ y ˜  l©©l˜ xxY‡4‰y v2z }–}„|‰‡8•x™D l ‡7€0l y–}B¨ V‡ y €7˜€ › ‡ ›2l x|w‡ ”y—x l4l—‰7€j€ –}—%wg‡V‡Vl”€ €££ ‡z | S˜ (l f,ƒ„  Px™)˜ lI¦ „ ¤ ‰w lux – l ›h—‰€ lhxl … !„  ©w˜) ›hˆ€l dƒ„g— u”y„}–†l ›„€ €7†– –Ú8xab`›!f¤„B −`<ê }– | b ˆ ‹ x5w | l Ÿ l €‚0–¤0 –}‡ ˜ ul ˜ 0l y 8„†„ l ‰— €7V‡ I¦ )€l0‡„ „› x – ‡„uw ‰— – l€7–}| › y7!ll ˆ€ ‰¥ l ˜ ¥l g‡Û s|‰¤–¤ 0‰w– V‡lz | ˆ€ ) g¦ }– }„ – ˜ A˜ A§ x l € {ƒz  © ¦ € ˜˜‡ ay”fDl u  —%‡ ˜ ¥ lP ˜ rl o› p‹ oz xxf”ysQllicI„ ix0o´l©‡ nu g| w.  q• w e| 0 – ‡Vz |¢d 0‡ y  ˜˜‡ ˜ l Ûg‰‰|f„ƒ– r˜l©x |‰}–˜'‡| lc07y –lDw‡zu e||ˆ€ –)¦g˜ • „lw ™l l | y”‡ zl©[|‘ua€0,8 ©b0¦g‡]–}y }„⇐– ˜˜ ⇒B ˜ ê ˆb b ‡#y ‡ d˜  ll €‚}– ©| 87y ›„Dl u –ˆ€l 8| „ y |g}– 0| – f= f | l ˜ — Œg—ae– 7€ ƒ„‡V— I— •‡Vw €ˆa|g0–}0 V‡ – ‡|Iz | l)x –}| 7y Dl u €ˆ) g¦ „ 2l l | s l wz | ˜l b§ x !l ˜ l y b f ™s ‡ z [a, „¿¦… 8†„ V‡ €u0 ‡› f b |‰‡ „ —%‡V€f b] fg¦ – l | bs ‘§ xl }„ „ƒ)›¨u ƒ„  aa a f(x) dx ˆintegral de f en [a, b] aEjemplos f(x) = 1 x – xP Q |‰‡ l©x }– | 7y lDu ˆ€ ) ¦g„ l`l |w|g–†| u“zw |w–†| y”l €7… 8 „}‡ 0 x – x P/ Q1.–  4 • w |g0 – ‡z |ˆ[a, b] ¿wg©l x s s § ˆb T P s(f, P) = 0 ⇒ f = 0 b S(f, P) = b − a ⇒ f = b − a  q• w |g0 – ƒ„‡z™| —If)(€xy )–¤0 =– ‡ z | l©x –}| 7y Dl u aˆ€ ) g¦ „ l—l | %s j Šˆˆ a l l d„ }– | ”y l ”€ …8}„ ‡ ™ 0—%V‡<€2.– Œ Pn 1 ˜ [a, ˜[a, b ]˜ a < b x b] xj = a+ j (b − a) , j = 0, 1, . . . , n . ™• w g| D – ‡z | f ©l vx ˜‰l ˆ € l D – l | y”l l | l n 7y l ”€ …¦8 †„ ‡ s „ ‰w ©l u ‡ „}– | b−a p 1 + 1 +...+ 1 − p 1 + 1 +...+ 1 n ¸ ¹S(f, Pn) − s(f, Pn) = a x1 xn−1 q x1 x2 bq = (b − a)2 , nab § f(x) … l €7–†Ûg©u„ƒ ˜ l gÛ |‰–¤0 – ‡z | j š ©¢ Š s — w‰)l x „ {¤z › ˆ1 = 0 n→∞ n Œ l  l | y ‡ g|  l)Dx se˜ l d w‰l € ˜ ‡!• dz 00‡}– „| 0 ‡jšh›  —‰Š s7€ V‡ ¦Iabdd0 x–x‡z 4| =ƒ„  se˜ l x ˆ Œ ‡ | ˜ lAa,b = −Ab,a Ax a—‰,b7€ ‡Vˆ g— Œ– – 0 <˜ l©xbê < a gÛ e| ©£›„‡ j –ƒŠ Aa,b + Ab,c = Aa,c l©˜ ˆT a, b, c > 0

104 CA´ LCULO INTEGRAL%—}„ ‡ ‡Vu q€ ©€}„5kjj{|z –}–}y †–}–l“„›x–†Š Š g¥ ¤–A=A©l s 1a¥sA,,a‡‘§s1bsb,lxx=„=l ˆ | — Aa,b T a, ˆb, s > 0 ˆ ƒ„l R„d 0•¤ w– w‰‡ z|glQ| 0 –A•‡ zw |1g|,el0o–}=‡g|ea18r„ˆit˜ ml„oy xhÌDi¼XÇ Ã½ˆÆ Ë ÇEÅ Ë ÃE©Ì ¼ià º “døÁ Ë SÅ Ì Ë Ï s7€„ I—}– w‡Vz)T €|‰€ˆa¤– ˜0 ,x‡h` b>—I„ƒ> ) 0ˆ€ l0 s (0 < a < b) Aww‰z ›1l),˜al jl“€‚–}+‡–†˜‰†– ŠqleA‰Û 81l©|g|,x€b–ªy7€ ll w ‡ n˜ e` p•ew rg| iaD–n‡ z o| Œ –l y – l |l b 1 1−1 Is j a < b ˆŠ ˆ  ab j ¿7 b − a¿ 7 a l P x2 ab 0< w |e„ I— ) € b −3.— ˆ = a dx = . . . , xn = b} = a, x1, y ¤– 0– ‡z |s w 8„}¤ w – l €ˆ ˜ l „¿–}| y7l ”€ ¦… )„}‡ a+b Œ {x0 j2[a, b] @H @H n 11 bb n 11 x2j−1 − xj2 xj2−1 − xj2 S(f, P) − s(f, P) = (xj − xj−1) ž −< P ž j=1 j=1 bb õ 1 − 1 , a2 b2 =P ö„ w‰w Dl ›hu ¢‡g—  „ V‡lI— €R)S€ˆ‡(4 yfˆ€,©P ε—%˜ )© −€ y7εlss>(fƒ„ , 0 ¿s x – Pε l©ux w e| ¨I— ) € y –¤0 – ‡z | y 8„“¤ ‰w kl b b ε a2b2 s“xl PxRε˜ )©l <x – u εw l)x ˆ Pε < b2 − a2 8„ ˜  ˜ 11 1 T j = 1, 2, . . . , n < <˜ 8| s x2j ‡ x˜ j™ x—Ij−© 1€ y x2j−1 y ¤– 0– ‡z | —I)€ˆ Pê s(f, P) = n1 n xj − xj−1 = 1−1 j=1 xj2 (xj − xj−1) < j=1 xj xj−1 ab n1 < j=1 x2j−1 (xj − xj−1) = S(f, P) ,„ ‰w Dl u ‡ ˆb dx = 1 − 1 x2 a b aC0 }– ‡o|nl©‹dx5i|¨cx wi—‰ogÛ €7nD–}›–el sl| €€”y sl)„ux‰w fiuI— c) )€iˆ€esqn›„¤t‡ew‰‰|s—l ‡ wdy ‡|ee|™i{ƒz n• w t§ g|e0g0–‡r‡ z |a| yb–†x|illwiu d– ˜–†a|  dy”˜‘Dl.u x ˆ€E‡)| j¦ges „ m)l ê p˜ lowse|  ˜ l–l „}¤„  Dx s D V‡ | ˜ –ª£ y Œ – ©l x 8 D ‡ y  ˜  § ›2V‡ | ‡z y ‡ e|  l| sl | ‡g|  l)x l©x –}| ”y Dl u €ˆ) g¦ „ lPl¨ |¥ rp7‡Vo`o[ˆ€ap8´,onDbl Òs|]±0iˆ Vrc©l7­ixY'²o´7y´©·nšlq Hç ©1j  .x‹ f ll ˆ€ l 8 ›h„XˆS—gŠ`q}„ ‡ )| ”y l €7–†‡V€ xl5¥ 8 | —I x ‡ x ¤ ‰w l [a, b] f ull „ ‡ | | Ÿ xDl u w – ˜ ™‡ }„ ‡ x ›„– x ›2‡ x

3.2 LA INTEGRAL DE RIEMANN 105 Œ w —%‡ | u ) ›2‡ x %— ‡V€ w l e|Ÿ l™h› I— —e) †„€ ‡‘y –¤0¤ –w‰‡z l|¢„ƒ„w •8 w „}¤|ew0–– ‡Vlz ˆ€|  f˜ ©ll x ˆ€ l 0 – l ¤ | w‰7y ll %s… xl l€‚–ªÛeε¤ w‰>l 0 gs §¢x l Pε = {x0, x1, . . . , xn} [a, b] bb ε P< . f(b) − f(a) l ˜ xw ¦g†– | y”l ”€ …8}„ ‡ x ‡ |[xj−1, xj] j Ûg|‰– y ‡ x Š § s€ƒH©l ‡x„›w „ ‡y Bê `| ©   Mj = f(xj) mj = f(xj−1) nnS(f, Pε) − s(f, Pε) = ¡ f(xj) − f(xj−1)k (xj − xj−1) < l P l ¡ f(xj) − f(xj−1)k j=1 j j=1 < ε f(b) − f(a)m = ε. f(b) − f(a)Py705tx¨ ll‡xDr‚pV‡| {dz8`oo€y „}l—p–})´8|!›‰w)onDw ‰|©lÒs0±xDÚ8ird˜sc) l‚­T“li| |B²o´ε)´sÔ ·nl—q[>‘|açl 2–,—0lHƒ.b€w ‡]xYSŒ|y2›ˆ –δ€ˆy ‡(f‡Œ ε”xx l©lDx[)ayxs >j,„}P0‡bs‡0x ]z|=lDjl©yy ¨B8x}– ˜{|y„x0€lw ¤l0‡ ˜›„w‰,›„x–}ll ‡I—1|s‰¥x8,x‡.M[– a.y xb‡.,jwPbsB,¦g§ b]§x–}smn|<ly7}x|jl δ)yw€”˜(¦‡…¦e|εl|gl) )!›„}„fs‰‡ )lI—ll‡ˆx |)|x f€y l¤y l©x ‡}–z f|¦g| ”y–}©l|l©‰˜x“u7y ll0€w €”©[|‰…¦a¦g–}8†•„,™l„†‡db‡ €‚]l›[|xˆ lj[−› a1V‡ ,l ,€b| xy7]ljˆl „ ‰w l )l x „¤– 0 x–w ] ‡g|  Mj − mj = f(yj) − f(zj) < b ε a T j = 1, . . . , n . −d ‰w lDu ‡h—I)ˆ€  w e|  P  Dx {z xl …h y”l |l )€ ê εn S(f, P) − s(f, P) < b − a (xj − xj−1) = ε . j=1רPPE§4l„}–˜˜˜¨¨ Dl||–lqljrrp‚p‚y7âox”xgÛe€`o`oloo0x[„ƒ|αm€”‡)p€ˆppl·¦…´88´l4xÄÒ|,„ƒHoo´B)nDDnpyβ)„ƒxÒ҄}zss‰µ–}‡l`¨‡±0±0–€]|iiou›´8„sdrdrlccw nRw•d‚­­‚liiw‡4–0B²²Boo´´–x|˜µ‰´))´lg|.‡··ynn——w—p7I¦|qqyDR‡¬‘˜e|çˆç7y$8–g|3‡l’4rD˜zhŸYl¥l)|.‡qh.‡Bl©x)Dr2‡ŠxŒw·g—j7€|}¬ |‰Œ–¤²ƒ˜„|e•g}„ƒ„wnw}–u–‡fl)¤²›g|Гn l8f…l©„ƒlcD|œfDlqo`xl„a‡–€ˆ–©ll›2‡‡l©z•m)–}z4xr¦›||w|xg¦¹‡ÒĔy|gl87pp„}–xDf©l˜lo||0D(sauó™sl‡–7yyt—ˆ€‡nr¦Dl8z)y)¶l·©l)|luÒi„ƒa| =¦e²ixYŠ€7€ˆ˜ýyD­y”‡„„ƒ)‡df´hl‹—leg¦ñg|ƒs—e€7−€l„lӁ¦…•‰wl©l|t|ï©l(Gw‡x©l2xxlw8X„[‰˜3 í[|)aaˆ©l|f„al™#yu x,˜=,[l)bxsBas‡b†–»˜xl|s],]€‡s²f”y–}ˆbb†–¦Ò2Š|lDsf€§wπw]xu7y)€ˆjz¹|4llD[ˆ€l©›2lqauº‚)xc„0—x€ˆwg¦,– –}ñ˜)lb|0„e„ y¦g–ll¤(j‡y”]−hms‰wDl„|Çðltl!uœŸl2ƒ„|I w€ˆu‰|s‰l© d)|[|‡“°[–}0¦g˜tαy| 0±,„‡[l€7ˆy7,la´xŸ–lDβ˜dlØ],˜ulj‚€i¬bs]FT|ˆ€tl0p7]©©¢§uo`¤–xˆ0¦eãlx n–}wgg„dã¦r)†–>‡ „}cxn ›2– )l“i˜˜s0o´x—–­©—©lxnˆw˜A´©0x„›z—‡qd|u¦e‰‡j|y€wD­xh‡l‡ryˆpÄvˆz|³¿}–x2r)l |y©¢oRxwÒYww˜ä8´r8˜|ln„sl

106 CA´ LCULO INTEGRAL“Ejemplo 5.  h • w |g0– ‡z |  x – x ± ‡z x – xw =0–†g¦ 0„,l ©l x †– | y”lDu ˆ€ )g¦ „ l 0 x – x –†”€ € ©l ˜ f(x) =  1 P/ , =p qql| sB§ ˆ1 f=0 [0, 1] rlx Œ22›– ˜x–,)‡l.€ˆεP€vx. l.>ly›„[,0g|‡ r,‡D˜01m1xV‡Rs]| „ƒ:=—Iy fwˆ€) ()x1€8x–€y}„¤u)}–¤ˆ wDw w−>–Œ–|e‡–lzll|2ε| h€ y7—%Phl—I© s )I—€ P)l„ƒy€)xy–¤€r0–¤xy‰|0–w‡}––z‡fyD‡¨›|z)–|‡%„…¦z5P| d¤xε–}s†–‰w|‰”€}{z„†˜l•‡…ƒ„ l8ll€72›§1ˆ}–[h‡V0lz€Hۉ,‹›„<1‰|r| ]‡–x›y‡'xy‡‡ƒ„80¤{zys€|ƒ–„€7{f‡N˜¤r˜ –w‰1us‰©,l 8˜l)r›„xSx2l ‡ „(0−‡s¿f}–x,|ˆlrP7y„€1$¥¤ lε,w‰Dl€”).‡¦…l48.<|„›) .©lŸ}„ ‡,w‡2xεrx| [êlm0y¤„5‡,w‰−10l˜]‡rêlT|mŸε—w−>w|1y|}0‡£ˆ%— ‡ ˜y‡x{ƒHr‡1|, x0 = 0, x1 = r1 − ε , x2 = r1 + ε ,... , x2m−1 = 1 − ε = 1 . 2h+1 2h+1 2h+m , x2m Œ –l y – l | ls § 0 ‡| l)Yx y ‡ y”l €7›2}– e| )›2‡ x ê 2m + ε (xj − xj−1)+ < 2ž S(f, N) = Mj(xj − xj−1) = jo †DB Ð †DB Ð †BDÐj=1 j m n o j im n o †BDÐ+ 1 ž (xj − xj−1) < ε ž 1+1 ž ε ž 2−h < ε. jo 2 1 − 1 2l | §`¥Ejercicio.  €7‡V¦I)€5¤ w‰l „ƒq• w |g0– ‡z | f(x) = x+1 x – 0 −< x −< 1, ©l x –†| ”y Dl u €ˆ)g¦ „ l [0, 2] 8}„ ƒ„ © € x w †– | ”y Dl u €ˆ8 „Xˆ 2−x x – 1 < x −< 2,S}– | u7y lDmu €ˆa8€s„l)xd˜ l —le| yw R) |I2› i™ e‡ m• xw e|a ¥0n–V‡n‡Vz€ˆ| q ê „ƒ ˜ l gÛ |‰¤– 0– V‡z |t‡V7€ – u –}|e)„ ˜ l žv– l ›!8 ‰| | ˜ l –†| y”Dl u €ˆ)g¦ –}„}– ˜  ˜4lll| y Œ  w b|e] ˆ •‘ w e||gD – ‡z | ˜ l gÛ |‰– ˜  sB!§ x”l  8w ¤„ e|¤ –w –I—l €)€ –ª£0– ‡z | ˜ l Pj  =x B‡ D{x–¤ 0˜ ,hx1 , Š. . . l©,xxn w} f suma de [a, b] f [a, P Riemann de 0‡|n ξj P [xj−1, xj] . σ(f, P) = f(ξj) ž (xj − xj−1), j=1j ƒ w 8| lv˜lwl E|˜‡ 8–}ƒ|σ}‡˜lj(–xf‡d=¤ ,xTw‰Pw xl“)ε¦gjl−−–}>'„| 1|”yU0sl w z”€„ƒƒS›¦… <8δσl „}€7‡>(ε‡fx ,0UI—DP)‡Vy)l)€ˆ”€8™x€l©„'l©xl¤ xy Œ  „ ƒ„  ˆbŠsuma de Cauchy ˜ l f l˜‰δl | l „s ‰w%—‡ llH´˜‡ım|x –˜ iσP–tl(e|l©f,xy7dP©l wex–)|es¿l˜€a)ll sI—qx P©s˜€ˆuyl m¤–DŒ0 ª–™l–€a‡ zs l)s| I—x dˆ˜ ) 7€le€ˆ–†¦ [Raly ,Isi‡ bel˜ m]|  0”y al‡ l |n|„ l˜nDb – DPl –| ‡b z ˜| <[}„x a‡l!x,ybξ– ]lj| ‡

3.2 LA INTEGRAL DE RIEMANN 107x –u ‰| –†gÛ ©  ˜ ‡ s U = p „ {¤z › ˆn jŒ l — w‰l©˜ l —‰7€ ‡V¦I)€h¤ ‰w `l x – ©l Yx 7y l p f(ξj) ž (xj − xj−1)i‰—l„l0j« }{nzš¨¤€„› ´Bthw‰w– xY®Ve–Šlª–7y”y7±¦…g¦I‹sllnˆ8rRp|q„Uall ¤²}„–l|•lq¢gwwDl„y7„d„×g¨ll©|g{}zv¶—‰„›enx–0ƒ„²¤xY€dqˆ–Dn–”y‡‡fz”yrDzsxl¦¨|l¬v’se}–0›„)lnw‡‰fUEx|©l‡©lP[xYxwxaz˜yy”y‰|Œ€ˆ€→,lliwn–¤‰—b‡d00D7€ƒ„t€]‡‰˜}–0lje|©lbˆ=4x}–l!›Šg‡vxu2›1r|xl©l aw‡€l©sIx b!›sxx€l˜y l©¢˜ˆl€|eãdg¦l‘xwãz0–jX›„ƒ˜–˜l™‡zˆ‡‘|ll| –}€7|l‡ ž€©ly”„ ©lxY–Uxly‰u ˆl›¨€0z|ˆk5)y)y„IwA–‰|‡˜%— !›| ‡ÙV‡ € ˜˜ ll žv– ll ›!| 8[|‰a|I, bŠ ]l ˆ | [‹a|,by ‡]g|  w ©l )xD|s ˜ ƒ„ ‡ k ‡– f ˜ x £)ž €”˜ h0l%¦ l § ‡ ý›„w ¨2‡¹ | § ¼¢˜ —%2) §‡V™ó €v' žvÇ k(% –“þl) s€!›”€ %¦¶8 ‡ |‰w | ÿ%¨ †²xsI±7‡xn©|lÒ Tl | eo[ar,ebm]sha§ „ ¤{z ›de Darboux.  H• w g| 0 – ‡z | ©l x –}| y7Dl u ˆ€ ) ¦g„ vl jXl | l „ xl | y–˜ ‡ ˜ l vž – l !› 8 |‰|IŠ n x – `§ x ‡z „}‡ x l©x d 0 ‡ y l| f – f  ˜  [a, b] s U p p →0 f(ξj) ž (xj − xj−1) = P xjl =| 1y – ˜–}| y7Dl u ˆ€ ) g¦ „ l„j l | l„ ‡ ˜ l k © €”%¦ ‡ w ¨ Š l| s § ˆb f=U [a, b]fξ§y§llx¨ –H}–|¢4|nDp‚f‡˜ |‰–,`o}„—Iy‡lz燩l‡ˆ€Ÿ7|)áxY´8Rxn©s €ysnD€syT˜)Òxsw–}zσ€±0l–8|eƒ{drz›PswVx˜™›h(­‚l„ƒl‘fB²s©%¦dzr´),•†·xR€σ—0V‡l©‡qP‡€s(xY€7xç“xnƒyfyf!›w xl)),(g¦– |x's€!P8˜S}–Pj|ln|o´˜)d4|n7y)l0˜–¤l−lo's‡|ƒ„‡”€¥)luly…¦x“fxs‡‘|[(8i¤xx˜ξ¤„„}xwgw„ƒj‡w)q−›!l©x ƒg¦1l‰sl¤€7−<x–},¤‚|–}x‰w|l“x›xwl—n7y—l j˜†–l8ll]y”I— }„x”€„lT€ˆlŠˆ–u–l)¦…u8|ˆξvžw€8l›l˜y|‰„†–†„P–}‡€ηl–lD‡ {¤uz!›¦s|n–[l‡yzx7y˜sg8|8ljξl|‰x„−s ©ln–|˜1l}„fx ‡l|,l˜¤x„˜x˜| [lw‰lja‡„x]¦ll¤„w,–%s„„}lufb¦g‡ ˜(x•]}–x©lwξw|ul xlg|›„n¦e7yy ˜|lDa–†))–|€”‡–™€x−‡¦…ny”|z›„|8ld˜f€7„†0‡(x‡lf…‡wηx „x8ƒy¦gn„}}„fx‡˜‡–})(‡'˜|xξl >0˜y7)—ƒ–llPƒl€”wnn−<˜…¦||‰| 8‡„s“ynj[„†x ‡a—‡lDd €x+xw‰¨,0 b‡©l–ll©–Yxƒfxu„y]xfylDs85w(–xulDx—–g8I—–|–j˜wg„‡|‰)z)©l˜‡|l©‡”€Dxˆƒ‡xx£ssss σ(f, Pn) − σ V (f, Pn) > b − a ,0 ‡| y €ˆq¤„  ¥ †– — ‡z 7y ©l x – xR˜ l—0l ¨ – Yx y”l e| 0–ƒ ˜ l „ {}z „› – ”y l —I)€ˆ™ ƒ„  xR”x w ›! vx ˜ l vž – l !› 8 ‰| |ˆ ‹ Œ|l x l©uw | ˜ ‡u„ ‰w u )¥ € s … ‡z l7y 8l©2›x – x‡ x ¤ w‰l f  w8 ›„„ ¤ g— ‰w „qll b„ƒ ˜ l <gÛ žv‰| δ–¤– l0 ⇒!›– ‡z |8 ƒ|‰σj| (š f)¢σ,ŠP(f‰˜),l−P}–)|Uˆ ”y lDƒ u<€ˆ)4ε¦g–}s„}†–‡£˜ ˜ ê Œ Ul   0Q —%‡V€ –†— ‡ y ˜  xw !›  P˜ lg¦ – ε> ε S δ> 0y b l | − ε < σ(f, P) < yU + s I— © 0€  4 4 , xn} 8„ ¤ w‰rl b b <δQ P = {x0, x1, . . . P {†z |g• y 8„ ¤ ‰w l ε ⇒ S uj P f(uj) < mj + 4(b − a) , mj = f(x) ⇒ S vj P [xj−1, xj] [xj−1, xj] ε [xj−1 ,xj ] f(vj) > Mj − 4(b − a) ; ”x w — y 8„¤ w‰l Mj = f(x) [xj−1 ,xj ]

108 CA´ LCULO INTEGRAL  n ε  , f(uj)(xj − xj−1) = σ1(f, P) < s(f, P) + 4 j=1 =⇒  nε f(vj)(xj − xj−1) = σ2(f, P) > S(f, P) − 4 . d‰w Dl u ‡ xl—y – l | l ê j=1 ˜ l ›„‡ ˜ ‡2¤ w‰lU − ε < σ1(f, P) − ε < s(f, P) −< S(f, P) < σ2(f, P) + ε < U+ ε, 2 4 4 2 S(f, P) − s(f, P) < ε , l‡ l˜‰| l y ‡k g| )  €”©l ¦%x ‡ w§ l©Rtx ˜‰†– | l ”y Dl u „l Xj l | l x xl—l |y y ˜ ¨ f ›  z ˆ€xD)s'¦gy „ 8 ™› e¦ – l z | „ – l – | Šˆˆ „ w‰lDu ‡ U − ε < s(f, P) −< b < ε 2 U+ , f −< S(f, P) 2 a ‘§ l©xYy ‡rrrrrbf − U rrrrr b a < ε, `T ε > 0 ⇒ f − U = 0 . a Si k ˜‡ scx l  Pε w |efI— )€ y }– D – ‡z | j ¦Û Ÿ” ˜ ©l ”x ‰˜ l d¤ w {†z Š ˜l [a, b] y )„ ¤ w‰l ›!08y ‰| 8|H„ ¤σ‰w(fl!, Qx – )Q… l©€7x–†eÛ w ©e|B ¨ê —I)€ y ¤– 0– ‡z | ˜l [a, b] y 8“„ ¤ w‰l ε>0 ε ˆ δžv>– l lS¥b Q(8 f„› ,b P‡<εx )δ¨−s'‰—y s‡€‚(d‡˜f¦%, P© x€uεw )¤›¨<wg l 3˜ S l s(f, Pε) − ε < σ(f, Q) < ε 3 S(f, Pε) + 3 ;— w‰)l x 0 ‡| ©lz Yx y ‡ ds x l …¨  7y l 1.| ,€ž,l.x€–.ln.¨›rrrrr}σ, Qm)(fg| x ,l|s'Q § ˜ )Ql0 −‡f|=lx ab–| {˜ yfl[rrrrr0a€ ,<l,y›„b1ε]‡ ,ˆxse. .§ƒ„ . —%V‡ –€ „}‡ y 8| y ‡ s ¤ w‰l b ©l xl „ „ {}z „› Œ –l y”ul ˜ l ƒ„  Rx xw ›! Rx ˜ l ,x yw m!› } ˜ l l „}– Ÿ”vž 8 –›„l ›!‡ xq8 |‰w || y j]PI— ε)=€ˆu {©x 0˜ , f P[yj−1, x1 ,.. Q j = a uj m σ(f, Q) = f(uj) ž (yj − yj−1) ; j=1

3.2 LA INTEGRAL DE RIEMANN 109x w —%˜y ‡ ‰˜| x uw ˜ 8g¦ l›„–}| „ƒ‡ y7`xl €”„}0V‡¦… ‡|) ›„}„u ‡ –@ ‡Qy7[wgx—‰˜‰i−€7–}˜1› l, f„xl ˆ€i› A] Dz0‡Vx ‡| |— ˜ l ¤–¤|0 ‰w –l ‡lz ‰||x©8 ‡„¤› ˜ ‰w lll |‰„}‡ƒ„‡ s xxq‰|x w‡Vw €7¦g‰|›!}– ‡|  y7‰˜l ˜”€l …¦l †„8‡ †„Qx‡ qxu©l j˜ x l D› P‡l|4ε‰| Q„†‡V‡!l€‘| ¤yw ‰w l l©„¤x›„© – – ‡V)e| „ s y l H s nn f(uj) ž (yj − yj−1) (yj − yj−1) σ(f, Q) =  ACBD1 E3F  CA DB 1 GE Fi=1 −< Mi ž j [xi−1,ø xi] i=1 j [xi−1¢,ø xi] uj uj nj bb n bb −< Mi (xi − xi−1) + 2 Q m −< S(f, Pε) + 2 Q Mi.j Œ lRl | ”y l | ˜ l € i–z=¤1w‰l Mies x€ ll ‰— y € l)xl | y  )„ xw ‰— € l „› ‡ ˜ l }„ ‡ x …8 }„ V‡ € ©l xf˜ l li=1 | ¢| [x‡ iy ,7€ x`‡ i−©1x ]bˆ ŠtR¥ ‡Vˆ€  l|l ε x }– E| › z f ‡©l Yx y  w z es x – n −< 0 – σ(f, Q) < S(f, Pε) + x ‹ s w 3 ˆ ˆ Mi h› g— „ l  w )| ˜ ‡b b ε Q< „ y }– ›!i =˜1l©x – uw 8 „ ˜  4˜ x l n k5l „› ‡ ˜ ¨‡ ) | 8z „}‡ u ‡ x l „}„ lDu „  6 Mi 1=1 bb n σ(f, Q) −> s(f, Pε) − 2 Q mi . i=1þá d ¤ w {z ds x – n sdx luy –l | l ε x }– s| › z x —ˆ á l| l „$‡ y 7€ ‡© x ‡ s „ƒq ›2– 3 ˆ mi −< 0 σ(f, Q) > s(f, Pε) − ε < xi=¨› 1 ˜ l©x – uw 8„ ˜  ˜tx l  w h› g— „ l  w 8 | ˜ ‡ bb n Q 6 mi wz › ‰w l 7€ – ˜xDsV‡ ls'| § Vu l ˜l ˆ€l 8›„„ ds‡ yxY€y )l€ˆxd 00 –‡‡z |¢| ˜ ¤–¤0‰w }– ‡©l ˜| q)l $x 0 x‡ V‡ ›h¦‰—g€ ls„ l0by QBˆ b l '„ | l €7‡€l¤ „ƒ | —‰€7V‡ —%‡V€‚0–}1‡=e| 18| δ>0l5k | )l x”[a˜ l, d ¤ ˆ w ™{z € l —‰€ ©l x l | y )€ l „› ‡ x —%‡VI€ t [a,b] ¤„ `0 „ƒ ”x l„˜ l „¤ x • w |e0 }– V‡ | l©x }– | 7y lDu €ˆ)¦g„ ©l x b]E|‰˜0‡‡l›„y cv¼8– rDl)2–i|gt‡$ótz ÚD|e' ‡‰— rsÇ €‚i(%l˜od‡Q„þul%— ldl‡©l| ex|xYw¤–y 0g|ix–n‡w0z xtƒ–| le}„ –†D˜ g‰¦x0‡h–r€7–u‡B‡az‡Vw|Š“b€7– ˆ–li€u l|©l –}iy7xe|dh—l 8a0l„X…d)yˆ lI— € d)es x €–xel‡8z s|„ªRh…l‘‡¤ilew‰‚|lml|¢ƒ„a—‰¢„}n‡¨€ ©l›n¤xl w‰l |„›ql y )‡Vx ›2l€7–ƒ€‡ l x™˜ j l 0€ ‡lvž |A– l  !›w§ e|wg8 q‰˜| |#—I)˜0€ l– 7yy ql¤„ !˜˜ Al¤ w‰¤„) l„ Û lPy ª– …¦r8o› plo|s”yicl ˆio´©l‹nxv|. wy Œ ‡e| lg|  —I©l fxD) sd€ y0f–}‡ DPy– ˜  l| s D‡ | x w ‰— € l § 2› ‡ !l {}z | Ûgsy ›„‡VS |ed‡  M § ym8 H„ €¤ )l w‰x —l¦sl ˆx £ – ‡z |t [˜‰al ,P = {xj}nj=0 [a, b] 0 l©>x 0 mk −> σ} , , l| ˜ b⇐] ⇒y 8„ T ¤ P b] σw‰ql >b b s >s'l 0| − [a, <d  (xk − xk−1) < s x – ‡ K = {k P {1, 2, . . . , n} : Mk kK

110 CA´ LCULO INTEGRAL ƒ„¤˜¨ q gw‡ 7p |lI—`o s˜)8´x€l –nDy M–¤ÒP00± –rV‡k=z7­ | '²§´©{·šPxq mjç ˆ}kjn=⇒x0‡|l)x€ƒ l ww„ x w ‰— € l 2› ‡ ql {}z | ۉ2› ‡ ˜ l l| l „ ”x w ¦g}– | ”y l €”… 8 „}‡ [xk, xk−1 ] ˜‰l f 8 „˜ ‡¤ xw‰rl σb 8„ |e) | ˜ —%‡ ©€fy )l x ‡ z–}|| 7y Dl˜ u l €ˆ)g¦ „ § s s>l |0y s‡ g|S d l©>x 0 y –¤0 – ¦l es ˜y [a, > b0 b] P <d n (Mj − mj) (xj − xj−1) < s σ . ™l 7€ ‡ js‰=x 1– K )l x€l „ 0 ‡|Ÿ w | y ‡ ˜ „l {}z | ˜ –¤ l©xR˜ l „ l | w |g0–ƒ ˜ ‡ s n„ w‰⇐j) lu=lD„ 1u©l¤ (‡xY‰wkMy rlk„  jb˜ }– K›h−P‡ (e—bxεmk„†<–¤>j©−)dd(00 xx–‰s j‡%skz l −|−x | l1s y 8x)‡S |j|g<d−σ1©l>s)xD=s−sg>0Dx k2‡–y („›8KbK„$ε−‡ (=¤ aMxwg)l l k>kIs¤ −‰wx P0– l m€P{§ ƒ{z1k=,s)2‰— (=,{x€7x.‡Vk.j¦I2}.−)(jn,=M€)xnˆ0ε−k}−)lm: xc1M))w >k−>e| u−σ0I— kmˆ ) €kKy ‡V(–¤−>x€c0–k‡Vƒ„2z  (|−b¥ε−˜‰x†– al—k)−‡ z[ay”1)l ),s x ,˜y –x b] ε Hƒ ‡ |4k}„ !‡ K (w x8 k„ s − xk−1) < 2(M − m) .  S(f, Pε) − s(f, Pε) = (Mk − mk)(xk − xk−1) + (Mj − mj)(xj − xj−1) kK εε j/K„ ‰w Dl u l©x y”l©u €0©g¦ „ < (M − m) ž 2(M − m) + 2(b − a) ž (b − a) = ε ,l t e— †„ ¤– ©)€ l ‡ f –†| ˆ ”y l y7l w ‡z | „ˆ€7–Ejercicio. 7€ †– ¨‡ 8 | 7€ }– V‡ –€ q„ƒq • |e0– x – s1 1 < x −< 1 xf(x) = n n+1 n – ˆ0 x=0c}–0˜ë« |‡rl ‡2©r7y|i·0 l¬itƒ”Ÿ–†8­€”e€v{$wf®V¦… rì|¤¬i8´iA´Pyw‰„†o‡‡ l p7xtƒHd˜q—I‡el[)la‘| …©€ˆ”€„}i,©r‡€0n8b±”x „t²iy]˜uDpe‡— h‡‡⇐µ‰g˜w xrV‡›r|⇒µ‰ay εp‚w‡zbDnx€§x€f>is l˜˜il©ž0l5d—lxxlway˜˜c…!›ˆ€d–}o–l©x €%¦n¥Dd'y V‡)„‡tjXzelŸYi|xn‡s0ly L‡©¢u‚€–}'y|E|–ãeadlzwÜlŸb0¡Šw–‰|e˜e|˜s¤–nsy0l —g4˜u‡‡ „}ucz‡¤˜j¦uae|…w‰lxsêlul©ˆif€7–˜ ”y„}t‡y™wgo–äs‰˜l d%—|©lw w§ oxvV‡lg¦ € ©l–x”mlpxxl €ˆbŸueyŠ›l q‡sHdn›„l tix—%|‰d—gol V‡‡V„}a‡€€5}„x„s0lDw¤ s™lu|I‰ws|s©l }„l x™‡„‡44xεR˜w¤ ˆˆ2l‰—¹w‰ 0Ç͉l)€7lò†–V‡x€ –¤¦I$s‡Vzw)|x €– 3U“þ l s „ „ –l l ll ˜ €

3.3 TEOREMAS DEL CA´ LCULO INTEGRAL 1113.3 TEOREMAS DEL CA´ LCULO INTEGRALPropiedades b´asicasŒ l  xl¢y – l | l §Proposicio´n 1.™ˆ áf P t [c, b] (a, b)Q cP f P t [a, b] ⇐⇒ f P t [a, c] b cb¨ p‚`o ´8nDÒ ±0rd‚­ ²B´)· q ˆ f = f + f.¥S(f, P) − s(f, P) k |e‡˜ ™ x‡ —Ixε)j€ a‰s xl aw ¥ |e[cd™ , blI— c|])ˆ€ w y Œe|¤– 0l–—– ‡yzI—|– l) |€˜ l y ˜)„l ¤ gw l cshP § §x ⇒ ˆ    yl –¤0[–a‡ z ,| [a, c] P†„ ‡xj P P ¯ [c, b] >0 P b] d< l ε| A w Py –}DP– ‡z¯ | [a, c‰˜ ]l B S(f, P) − s(f, P) = j S(f|[a,c] , A) − s(f|[a,c] , A)m j Sy –ª(h… fB‡|[Šc,˜b]l Bl)x −l €Rs› (fl |[‰| c‡V,b5€ ] ¤ + , § Qε I— )€ y ¤– 0 – , ‰wBl)m < ε, w‰lDu ‡2)  ˜ ™I— )€ lz | y”©l x – x—j ¤ w‰—l )l x |‰‡„| ©l u  ‡Vz | „ ⇐w‰l k ˜‡ ε> €s xl 8 | Pε I— ) € y –¤0 – ‡z | ˜l ¦ ˜εl ˆ y 8„ l©x ‹¤ |y ‡ S|g( f©l |[xDas ,c] , Pε 0 ε § [a, c] [c, b] < ε ˆ 2 2 ) − s(f|[a,c] , Pε) < S V (f|[c,b] , Qε) − s V (f|[c,b] , Qε) „ w‰lDu ‡ (S +l©x SV )y7−lDu (€ˆs) ¦g+„ ™l slV ) = S V V (f, Pε n Qε) − s V V (f, Pε n Qε) < ε , f }– | | ˆ[a, b]á u w bs ¤ ‰w „l ©l ™x l „ l w z |‰–¤0 ‡t| w z › x l €7t‡ 0‡›h‰— € l| ˜ –˜ ‡ l | y€ l „¤ x ©l xS— V Vl § y „ƒ x sl V |V s©l x — w‰)l x– 8 a$„ fu„ƒ x w !›  ˜ }„ ‡ xfw z ‰| –¤0 ‡ | w z › l €7‡ x D ‡›h—‰€ l | ˜ –˜ ‡ ©x s €  –†…8› ”y ¦l es l | y€l ƒ„  x § s §`l | y € l ƒ„  x SV § „ƒ x s ¤ w‰lx ‡V| c§ ˆb f f S sV ac  w 8 | ˜ Rt ¥ V‡ ˆ€  s a0}–]|‡s¿y7|‚xDl l¨u „ƒˆ€x dx– u0 –dw‰‡ z le| fi꤄ ‘ni–cu iw o8n„ ˜es˜‰a˜al § €0)H„ € l©x§ ”€ P ¦… )t „}‡ [b˜ ,l f  =˜ – y 0–ª… – ˜  b l =„ƒ‘ −}– | a ‡ l by ‡ <˜ la„–}| — 7y fl ‰˜a f˜ b”y l©fu b cb f= f+ fI—–}| )y7€ˆl €”…¦y) ‡a}„ ‡˜ x‡ x wg}„ ‡l ax •ƒ‡V| €7w z¨› › )cl|7€ ˆ‡ x s  w 8| ˜ ‡ f xl ¢}– | 7y Dl u €ˆ) ¦g„ 4l l | l „v!›  § V‡ € ‰˜ l „}‡ x ¤ a, b, cj –ƒŠProposicio´n 2. Œ w %— ‡| l €5¤ w‰l f, g P t 's § 5 ˆ Œl … l 7€ †– gÛ © 8 |ê f+g P t [a, b] § b [a, b] λ P (f + g) = ˆb b f+ g aa a bb (λf) = λ f j –†–†Š § ˆλf P t [a, b] aa

112 CA´ LCULO INTEGRAL j –}–}†– Š Œ– l |f −< g s l | y ‡|g l©x ˆb b f −< g [a, b] j –†…gŠ §ƒ ƒ rrrrr rrrrr a a ˆf P t [a, b] b f −< bƒ ƒ a f a j …eŠ § @ H @ H jfg P t [a, b]ŠˆˆSchwarz rrrrr b fg rrrrr −< b 1 1 a 2 f2 b2 ž a g2 Desigualdad de a j… –ƒŠ Œ– 1 ©l Yx ‰y „z d 0‡ y  ˜  l| ‰s l | y ‡ g|  ©l x f P t ˆ[a, b] l 7p ˆ€ o`)| ´8˜nDÒ‡ 0± rVx w­7'²!›´©·gq  ˆRx j˜ †– lŠ s ž€j –}–ƒ–l Š ›!§f8 ‰|j –†|}– –†ˆŠ [a, b] g g ˜¨ j 0‡| x – ˜ l ˆ€ )—€ d¤ w {z − fŠH€ l)xw „ y 8 | x l |g0 }– „}„ƒ x 0 ‡| x –ª£f—%xt gV‡ [P €aP ,±x”5kjjbtle…–†Šˆg…€]l©ˆŠ[x€Ša§h–Hƒu,áf2fbw‡  ¤„›2]8‰˜|‰!ˆ„x©l‡˜‡lq0x  –‡x¥fu4˜ l›hgqw ¨ ˜—%)…=l–†„‡–|˜ Yxx y”y41¤– lD0‡`˜tuj– (‡ˆ€z¤xf|)8‰w ¦g+„l™l˜„ lƒlqg˜fs )lwƒ—%2e|P)V‡ −„—g€ t †„ •¤–(l w©[fŸ a)l|g−,›„€ 0b–jg—g‡]z–}„}}–|f)‡†–j2Š—%0m fV‡l‡s(€–||x§ y)0−}––h|l=wƒ€fyP ‡s¤ x‰w – l — gw ©l þ±˜ ql § x l €–¤ ‰w l ƒ ƒ xP – Œ  ¥ v )7€šÚ¦Tê λ P 5 xl—y – l | ls %— ‡d€ ˆxƒ ƒP/ −ƒ 1 f(x) = f f 1 −< f −< s t§t w[ae| ,hb]–}| ⇒7y lDu hˆ€ )2¦g„P ls t € ©l[ax w, b„ y ] j }– –}ƒ– Š s @ H @ H@ H b b bb 0 −< (λg − f)2 = λ2 g2 − 2λ fg + f2 , a a aay„ w‰–†…¦Dl ‡‰u ˆ‡ l„ ˜ – x ˆ‚€ †– 2› †– |I)| ”y l˜ –l ©l xYy  lw d0– ‡z | ˜ l–x l©u w | ˜ ‡ u ˆ€  ˜ ‡ l| ˜ l¦ —l xl v€ | Dl u )£ λ ‰w lDu j … –ƒŠ  ‡V€ ¥ †– — ‡z y7©l x – xDs8lD¨ – Yx ”y l |™D V‡ | xYy 8| 7y l©x —%‡ x– y †– ¦…  x c § C y )„ l©x ©¤ g— w‰}„ l–¤© c)€ <j ƒƒ < CQ j 0‡ „› I— ‡ x ¤– 0– ‡z | ˜ l 0 ‡| y }– | w  l †– | ”y Dl u €ˆ) ¦g„ l Š 's § „ ‡ 1 t [a, b] g… gŠˆˆ P gs's l |Ejercicio. y  €7‡V¦I) €‘¤ ‰w El x – l©x 0‡ | y }– | w  l| f(x) −> 0 —I)€ˆ y‡ ˜ ‡ § b ‡V|e l©x f(x) = y‡ ˜ ‡ xP [a, b] f=0 0 f ˆ[a, b] x I— )€ˆ aTeorema fundamental del c´alculo. Regla de Barrow(‹ T| yF‡C|g)l©x êTeorema. Œ l  fP t [a, b] sds§ x l  F(x) = x —I) €ˆ¨ ©  ˜  xP ˆ[a, b] ¡ Ë ÅÅË Ç Ãº f f(x) a f# # # (x) = j †– Š FŒ l)x ˜0 ‡l ›| y ˆ}–z |x w  l | 0‡[a| ,y b†– | ]wˆ l| [a, b] s‰l | y ‡|g ©l x l)vx ˜ l ‚€ –ª¦… ) g¦ „ l—l | 1 j }– –†Š –  l©x§ f  x0 P F x0  1 + (A 687 . 2x FV (x0) = f(x0)

3.3 TEOREMAS DEL CA´ LCULO INTEGRAL 113 ¨ ‚p o`´8nDÒ 0± dr ‚­ ²B)´ · q ˆ j ƒ– Š Œw %— V‡ | u 82› ‡ x¦ƒ ƒ −< M l | ˆ[a, b] f(x) x −(x − a) ž M −< f −< (x − a) ž M x –↓ a „ {¤z ›0 −< x → a+ ↓ –xll →|| laa+ˆ F(x) −< 0 „ ‰w lDu ‡ Œ ©l x 0 ‡| y }– | w ™%— V‡ s€R„ƒ ˜ l€l x —l¥ y lF8| y, z P (a, b] y < zQ ƒƒ = rrrr z f rrrr −< zƒ ƒ F(z) − F(y) f −< M(z − y) , „ ‰w lDu ‡ l©xvw |‰–}•ƒV‡ €7› l › l | yy”l 0‡ | y y w  's §`l ‘| I— ) € y ¤–  w ƒ„ ) € ©l x 0‡| y –}| w  s'l | l ˆ(a, b] j †– †– Š ¥$l F)›2‡ x ¤ w‰l F+V (x0) = j 8 | )z }„ ‡ u 8› l | y”ls f(x0) Šˆˆ Œ  h> }– | F−V (x0) = ê f(x0) 0 @H F(x0 + h) − F(x0) − 1 x0 +h h f(x0) = h dx . x0 ¡ f(x) − f(x0)k lRt | ¥ y V‡‡ ˆ€|g  s l)¦0x ‡ ƒ ›„‡ )l x 0‡| y –}| w  l „ | w‰lDxu 0‡ ses‰˜x –  ˜‡ ‰w l™x – ƒ − ƒ < δ s s y 8 „$¤ε > 0 S δ > 0 f ) ƒ < ε Q 0 x x0 <h<δ f(x) − f(x0 rrrr F(x0 + h) − F(x0) − f(x0) rrrr −< 1 x0+h ƒ f(x) − f(x0) dx ƒ < εh = ε. h h h x0 C‰—j –†€7†–o–}Š ›„nŒ s– –ye–ªg…¦cul©evx˜‰nl‰˜ clfi€7a–†l …s| ).g¦ [aj„ –†l,Š ‰sbŒš ]›– ˆ ©l x 0‡| y –}| w  l| s „ƒ„• w g| 0 – ‡z | F(x) = x l©x5w e|  § f f [a, b] [a, b] (g) a I ©l x 0‡ | y }– | w  l| s f [a, b] @H d g(x) j¡ Ë Å Å Ë Ç f# (2) à º dx a f(t) dt = f g(x)mΞ g V (x) . f(x) = ¨ p‚`o 8´ nDÒ ±0rd­‚²B)´ · q ˆ j }– –†Š Œ l  g(x) j se˜ ‡ | ˜ l ˆy t g— „}¤– ©) € „ƒ™€ Dl u „ƒ ˜ l „ƒu © ˜ l |eB ˆ f x3−4 x dt H(x) = f = F g(x)m F(y) = . a 2x 1 + t a Regla d| ey ‡g|B al)rDx rs owb f (I). Œ l  f D‡V| y –}| w  l| § G w e| 2 —g€7–†2› – y †– ¦…  ˜l fl| [a, b] Q − l = G(b) G(a) ˆ [a, b] ¨ ‚p o`´8Dn Ò 0± dr ­‚²B´)· q a f|g©l0 xƒ– 8 ‡ „ y y †– ¦…  ˜ l l § Dx {z s —%V‡ € w |I 0‡| xl  ‰w l g| 0–ƒ “‰w l©x s ˜ ˆ[a, b] ˆ l„ j c¥ F(œ x)Š =˜ –}• x ˆ€ R—‰€7}– „› – G(x) f | —%[© aˆ€ , by]‡ Q  x P ˜ l a€ l F(x) = +C ‡

114 CA´ LCULO INTEGRAL )€ˆ x = a € l)xw „ y  C = −G(a) Q „ ‰w lDu ‡ b F(b) = f = G(b) − G(a) . aRegla dy –†e¦…  B˜ al rrfol w| Œ l  fy V‡ P |et ©l Dx[as , b] §q”x w —%‡ | u )›2‡ x ¤ w‰“l l0¨ – Yx 7y lHw I| € • w |g0– ‡z | ‹ | ˆF(b) − F(a)F —g7€ †– ›2– (II). b bb¨)—g7p „}`o–¤© ´88 nD| Ҙ ±0‡Vr 7­ l²'„´©· q j ˆ ¥cŒ œl ˆ[a, b] f= f(x) dx = dF w g¦ |e–†|4y”lI— €”)… € 8y }„¤– ‡0–s ‡z |‡ w w l ˜l  Š P˜ –†•=l € a 8 ¤„ ¤ – ˆ€  [a, b] Q aa l{xg| 00 ,ƒ– 8x„ 1l,|t. .©. ,˜ xn}xw = F(b) − F(a) É ö ¼Xªº ƂDÀ º æXÐ nj n F(b) − F(a) = F(xj) − F(xj−1)m = f(ξj)(xj − xj−1) ,SC„}˜ ‡E(l of7€ ¤,nª– …¦‰wPs)l!)e¦gc§l©„ l©u™x xvfew lnl©|e|xc i–}[a|axsy”w,lD.b!› u ]€ˆj)§ –ƒjg¦Š˜‰=u„ l1lj Vê‰s,vž vx”V‡ z l–€7VUl › ›!xP – wu8t „ƒw‰‰| [l|a˜¿sƒ„,ulb˜ D]–iu‰s‡Vn8x|et—l„› 0e„yg‡gw – rxlx a–| σ‡ zcl |j(i=fo´ˆ ,1nP)pfˆ oƒHr‡V›2pa‡ rtse(fs,bˆ PŠ ) Œ −< σ(f, P) −< – u, v x ‡| bb uv V + u V v = u(b)v(b) − u(a)v(a).2›‹j –}ƒ–|V‡Š y| ‡j ‡êzg| ya‡d‡z l©€‚e| Dx› s s¿w „ƒ˜  l 7€˜ a–†l…„)c¦ga„ l!m§ biDo‡| de˜ l v€‚aª– …¦r i˜ablue ˆbŠ Œ l  [au,:b[a] Q, bxl]  → [αD ‡,| βy ]–}| l)wYx y  €7¤– l y 8 › l | y”l ˆ f | VRP t [a, b] β bj f(x) dx = f u(t)m u V (t) dt .¨Œ l p7`o 8´ αDn Ò 0± rV­7i²¤©´ôç dq ‚pj n –†Ša   x • w g| 0 –}‡| ©l x u § s v„ w‰x Dl‡ u ˜| ¢‡ 0—%‡‡V| h€y †– ¤„| 4w € Dlx©us ƒ„ „ w‰lD˜ u 4l ‰‡ Ù †– |)€”y”7€l©‡u 0€ ©¦gj „š l©š xŠ sˆ  g = uv Q g [a, V = u V v + uv ÎV P t b] ˆb g V = g(b) − g(a)j }–a–ƒŠ  )€ˆ¢ ©  ˜  s“xl  F(y) = y f Q ) —g„}}– ))| ˜ ‡ j dê ƒvŠ s“x`l y – l | l ¤ w‰lFV (y) =)€ˆfs (y) ) ˆ˜  ˜ ƒ„  ˜ l ƒ„ A )  ‰˜ l e|  s y P [α, β] j αg V (t) = f u(t)m ˆ[at , Rs x l lD=u ƒ„  j uql (tÙ ))m €”ˆ €7‡   V‡ j€`š š„ƒŠf s € lDu t P bg— ]†„ –¤©8| „‡ g(„ƒqt)€ F u ˜ V (t) bj b f u(t)m u V (t) dt = g V (t)dt = g(b) − g(a) aa jjβ = F u(b)m − F u(a)m = f(x) dx . α

3.3 TEOREMAS DEL CA´ LCULO INTEGRAL 115 Teoremas del valor Œm– efdl©ix o0 p‡ |ayr}– a| w intl e| g[raa,lbe]ssgl | y ‡|g ©l xvDl ¨ – Yx ”y l [a, b] y ) „ ¤ ‰w l ˆ(TVM1) Teorema. cP 1 b f = f(c) …¦x¨ l) p‚}„ V‡`oV‡ ¦€ b´8l©y nDx–−lÒ ±0›| dra¦l }{z­‚s|‰²B)´–}–}a·|„›—q y7ç ‡lDuŒ ˜l)z x‡¨ xYs y €8 l©| x ”y— ¦ls l 's  ly „†– …”y 8l ›‡V€ ll |›!7y l se©l ˜ “x l y f€7†– …'l |–ƒ8X„[ˆa,Œ b–d]|‰ˆ ‡ k5s'xl l § x §€ˆ– 8›f| –†0l›2‡| |‡ 8 | M „}‡ [a, b] m m −< f(x) −< M b 1b c(§!u| T€ˆl P˜)ˆ€ V¦gl8(„„m‹aM¤„l—s‰– xc,˜(lb—‰1blx| )€7G„l −V‡g|[¤la—g)0|w‰a,–}– l—l©T}„b))l}„˜‡`e]Yxl <”y„s‰o)4˜l lr8”€ |e)|€j l©ym”ya‡uxl c¥ ‡„ƒ|g7€fa)†–ˆš.‡V€Š©l<€ x€©lŒ˜ ©lxY–l0lMyvx ¨fƒ„ l– (xYl©„˜‰xby”x l©l• −›„w0cx‡g| ‡‡a|0PYx )y}–gy ‡–}[€ˆ(|a|dxw©l,0 )xb–u ‡V=z]0|¢l ‡y| 1|8I— „[y—%)ma–†¤€ˆ|©,q‰wˆ€wb<l—‰]x“€7ygs V‡‡xb§b¦I˜cl f)−g‡ xg€– xu)la=w w‰x êt| l f|‰a(€„ƒ‡¨cl©f)x| 0w Dl‡ <„bu|gy g0y˜„M–†gwˆ‡!…¦x  –,› ‡ z l|  z s D ‡| £ xRul £x¦ÇX½ˆÆ Ë Ç –}| ”y l 1 (d‹ Ti| oyV‡˜‹|gMl „g „ l©2€ Dxl©))sxz „¤w T „w ye}„o2‡ ˜ r‡}–e| mxy7–Dl u au w €ˆ.–8l X„j| ê 7y cl x”l5xwgl „l „}„ƒ8 !› )€ ˆ € a a valor me- −< †– Š Œ l  f A§ ˜ l −> segundo teorema del ˆP t [a, b] 72 0 l D– l | 7y „l l | [a, b] Q x l g 1 x6 dx bc —I) €ˆh8„ u“z w | c P [a, b] . 0 1 + x2 fg = f(a) g 1xhXÇ ½ˆÆ Ë Ç −< 7 3 aa 8 −< j á x – f −> 0 § ˆ€ l 0– l | y7l s S c P [a, b] y )„ ¤ wgl bˆ Šb b 1 j †– †– Š ©l x ‡z y ‡ e|  )l x fg = f(b) g § 7y Dl u ˆ€ )¦g„ l sgx l–y – la| l 2 1 − x Œ – f „› ‡ | g –}| c dx b cb I— )€ˆh)„ “u z w | c P [a, b] . 0 1 + x 3 fg = f(a) g + f(b) gÉ b û º’‘ Ë D§ ý Ð −< 4 ˜˜ `ll 7Ÿ ý )˜ 2› l ñ a„› ‡ Ó x‡ï Yxl3yí €ˆwd0g| –0‡z–ƒ| ˜ ˜ ‡jdql ˆahǓ!©l ¿ˆYx 7ygw l ©l ˜€ l©2l x wl |g„ y 0 c‡˜ | 2‡ y ˆ€€ )l €¤ x wl!– lw € ql w ‰˜| l ˜ 0l y Yx)y}„ ˆ„€ ™l8 Dy'–l‡zz0| |‰–¤l0|‘‡ l¤ f„‰w ql}„ –†¦‰x 7€‡VE‡„ƒ80› – yl | y”lx¦ÇXˆ½ Æ Ë Ç e|  „› ‡ ‡ # » º‚ñ(687rrrrπA x x dx rrrr −< 2 | ˜ 1 l 4§ ˜ l 0€ l 0 – l | y7ll xl l „¤A • g| 0– ‡z | j L‹xl e| 8 my| ‡ g|am d©l xDe§ s Œ  f −>`x 0˜ w | [ay –},| bw ] Q Š  g P l t [a, b] Q AMbe}„ ‡ l–x B…¦8o„†n‡Vn€ l)e!x t.lDB¨ y l 0‡ | € ›„‡ x | ˆ[a, b] g a b m f(a) −< fg −< M f(a) . a

116 CA´ LCULO INTEGRAL¨ 7p `o ´8Dn Ò 0± rV7­ ²'´©· Âq µ‰p h å@v Ñtw i”˜ç j †– Š t g— †„ ¤– ©8| ˜ ‡ l „v„ l !›  s b = µ f(a) 0‡ | µP[m, M] Q  ¥ ‡V€ˆ s —%‡V€ x l x g 0‡ | y –†| w  s § )—g„†–¤©8| ˜ ‡ j [a, b] fg € 5¥aš Š ‰s 0l ¨ – xY”y ql w | c Py ) „¤ w‰l ˆc a g µ=j }–gw©lƒ– Šul— Œl0y w¨–ª…¦I—– Yx ‡y7q§l| u ˜w 8l| „› 0a€c‡ l x 0– ¤l w‰| 7yl Hl fy 8xy q›l) „g¦ ¤˜– l wgzl | 0l Q€ l)0—g– }„l –¤|©8y”l| Q˜ l‡c| „ƒy€ ‡ I—|g) l©€ y”x 5l ¤„ 2j ƒ– Š• w ˜ |glH0 ©l– ‡xYz y7| lff7y (l xV‡ )€ l −¨› f (sbx)l“©lx x– u |‰‰w ‡l|¤ P [a, b] bj j c f(x) − f(b)m g(x) dx = f(a) − f(b)m g(x) dx ;„ w‰Dl u ‡ a a b bj c cb fg = f(b) g + f(a) − f(b)m g = f(a) g + f(b) g . aa a ac3lô§„wƒu©l)zax€€ˆ|e.x”ysl8)z4wg„€0xl©d˜lx™‡5Vkfx0–}e˜€”‡„‡©l˜€fil˜x|l©lA%—n)2lƒ„xÛg˜ ‡”€—%i‰|}–7€$xcwPYxD‡‡–7yi–†e|x˜|l‡}„owLƒ„7€˜|n)—}–xD–€)lV‡Ixell¦slHxw€)|Cs€”‰˜›„ˆÛe—˜7yl‘˜©l0‡l5lAx©–€”lxz©lxgۀ„ •™xwClD‡V)z0z¥˜–€7€¦l'…“xI‡l›“Vls–ˆ€O˜B˜xwl¨‡lˆz|‰„¤„}N˜!„}€–‡x˜l©xxˆ0‡8Ew–}™xx™q››|‹eÛ S˜‰y‰˜¦g0|0l‡zll–y}– lx‡Û‰‡€lG| |‰™s—gl˜„“y7h…–¤„¤‡˜©l©0E8‡Dxxl–}|‰„‡s¦xwO‡—‰|‡¤l0Dx€l©‰w|–sdM˜‡zqxzl„}|l‚)l‡yu—‰x|ˆ¤–l)E—0€´zu|Œ€l‡l–lDwlTx™€ˆ”y –0€wg)—%x˜x)u€˜)R‡„l€l©)lg|—lqxx“˜ƒ„I„›8x8˜€Cz˜|ell}„‡ll™€„›DxwA)Dxs}„‰—‡†„s‚€‡VI—‡€0x|S¤„l©s)‡luxˆ€x›q|l–§– 7y| •l)wg„ƒ¦…ƒ„V‡yz˜c˜d‡€7x –ƒ›˜„s¦…)w• z)w|h…w› „}—x”y|e‡–„ƒl˜„l0lwqx|l}–|–}V‡›Ú|l©x| x¿7yw‡l˜©ll |||§lx£C´alculo de ´areas planas([… aal ,€)by }–t]()z ˆI)€ )„l ©l  xRyxe‡ c=y 8in„ atAo§ s„}x†– l›2i=m– y bit˜ ˆa¨ dŒ —%wo—%V‡s€‡p| w o˜ e| r€ `l c›„ uw‡ r”€x v…¦¤a wgsyl ey=n(xey)x(xl)p)“x l´ısw cl|ei„t™al •sŸ w l |gX0 – V‡Xz | sds 0§ ‡|„ƒ y x € w l  y l | x –}| A = b rr y(x) rr dx .lŒ |– s l%— d„ V‡ }–€| y7l lŸ l”€ …¦h› 8a—g}„ ‡„}‡ (s ay,(ba))s‰>l „ 0)z €l l y(y x‡ )y 8=„ x”0l y {ƒ–z l | lhjx ‡ „ƒ8 › l| y7l Š“„¤ xRx ‡ „ w D †– ‡| )l x x1 < x2 € x1 x2 b A = y(x) dx − y(x) dx + y(x) dx . a x1 x2

3.4 APLICACIONES GEOME´TRICAS 117(§0b‡ „ƒ|) y x tz € € l l   y€x y l ‡x |y …8 [l„a€ A,y –¤b© }„]8}– „›ˆ„ ©l –x y ˜ – —%‡V€§ ˜‡ x wb€”ˆ¦… Œ fx lq‰˜ ”x Rlw —%l ‡ w | dl 0 }–¤ ‡Vw‰| l l©x y =l yy21(x x‡)E| s •yw |e=0 }–y‡V2| (l©xxD)s –}| y1 w x =a x = A = b rr y1(x) − y2(x) rr dx .(l¥ c|‡V)7€ [}–cDÚt ,z‡ d€| l ]y  8ˆ „ y )l ‡x ay 8 „ A }„ –}l„› – y ˜ d` ˆ—%V‡Œ € w %—w ‡|e|` ˜ € wl 2›”€ ¦… ‡ x x¤ w‰=l x(y) s™l©xvl „ w l|eŸ ql •Yw Vg|YD –s‡z §| sƒ„ 0 x‡ | € yl –}| yw x  y =c y = x(y) A = d rr x(y) rr dy . cE€ lDju e–}r‡c| i©l cxRio˜ sfl .)z € 1l .–x  l©¨ x — I— l ) € y )z–†¦%… ‡ xƒ„  2˜π—Iy2)2+€ D)=z‡34%¦ E| ‡2§ ƒ„†•xV‡6x 7€ π˜!›y–†−2'… – ˜l ˆ 8„“ ª{z ‚€  w „}‡ x2 + y2 −< 8 l |˜ ‡x2›„l©x.}––|e1‹ ˆ˜ „ ™ )z %—€ l‡V€5˜ƒ„  €  s w  ˜ €‚†– „ z y”l 7€ s‘‡  w 7€ …'}– „ ˜ 0l 7y l l vx ƒ„u„ € )€z ©leÛ u© – ‡zvx A| ˜ dl 0ƒ„‡  y 4  {}z |l l ‡ ”€ £ x x y2 = 2 x y = x2 =2 3˜ l y x2 = 6 t (==z I€ Ilxy)!((ttR„}))–†2› ec– yiƒ„ n ˜fxt`ol s Dw ‡l8i| mD†– l‡i„t| al)l Ÿdx l oI— Xs) €ˆV pX8›o—%rlDz‡Vy c€7€ u¤– ©w r|QvxHa)˜ s€‚l50 e‡w n|ex —}–p„› ag—w r€”„al…¦mˆ ´e5kγtDl¦sry !§i8 c„}xaƒ„ w s˜%— 8‡›| l | 7y ls ”x l 8| u 8 „› ‡ x ¤ w‰l(a) x l y(t) x ‡ | ˜ l „ƒh0„ƒ x l s y s x(t) • [t0, t1 ] ” (1) j [t0, t1 ]m • x V (t) > 0 T t P ]˜ ˆ™%— ‡d€‹ l| € y y •‡¤– ©g| 8y „(l©©l txDx )sBxl>„ =0)z € T t P [§}„t–†02› ,–ty 1 x(t1) l)xl „ ©‚€ D ‡ ˜ l „ƒ™ w €”…¦ γ Bs l „ lŸl § ¤„  x €lyx… x= l  A XV X x(t0) t1 A = y(t) x V (t) dt . t0 t z € l ‰„}–}„› – y  ˜ ‰%— ‡V€¢) €‚0‡ x w  l €”ˆ€  ˜ ‡ x ˆ Œ l 8 | ‰w l x = x(t) ƒ„  x‘l  w d 0–}‡ | ©l x(I— b)ˆ€)8 › 0lz y 7€ ¤– © xR˜ l—w |eu  €”…¦ γ Bs §4x w %— ‡| u 8„› ‡ x ¤ y = y(t) ‡| ˜ „ƒh0 ƒ„  xl s” j • lx l (1) [t0, t1 ]m x(t) y(t) ˆj j • x(t0), y(t0)m = x(t1), y(t1)m ˆ• x V (t)2 + y V (t)2 =W 0 T t P [t0, t1 ]

118 CA´ LCULO INTEGRAL‹x¤ lwg|| ¨lyy ‡– ©l˜g| x¨‡ ©l—%8xD‡|s xyl – y„ –†¦…)V‡z €ˆ€ l©qx€‚–†– ‡‡Vs 7€s8– )lH„l x| € yl D ˜d‡ 7€ € Al € l |gl „“l 7€ –}€ˆ|  7y ˜l ”€ …¦—%8V‡„†‡ € l ™„ © €‚D‡ ˜ „ƒtl „ƒ uw ”€ …¦w  ”€ …©l  x γ€ l l0 |‡V”€ [7€ t– 0˜ , lt1 ]| s – ¥ t [t0, t1 ] t1 t1 A = − y(t) x V (t) dt = x(t) y V (t) dt t0 t0 1 t1 j = x(t) y V (t) − y(t) x V (t)m dt . 2 t0E•x l– lj€ e„›l |gr‰—cD €¤–ilcil‰˜o|™ls.xw w 1|E.—g– „ƒ†{z8€‚ |‰ w‡ „}cs ‡ iuscl†„ o‡2w i)„¤d)| e€˜ u ‡ l©x l©˜ ¤„xq—l”y l w w e| {ª”€zq€ˆ…  u„w{}z |„}¤ 2‡lw‰™€cl wg€y ©llˆ€ ˜)yÚ8B xˆ w ¨| ˜ — )l wx „}| †– Ú8y ‡u© € Ûs Ÿ›!‡ 8˜ | l y”lƒ„ q‰| –0lz†–|€‚˜ w ‡ |x l £ ‡  }– | ‹ „ )z € l 2D ‡ h› —‰€ l |˜ – ˜  l| y€l „¤2 D–}D „†‡ – ˜ l x = a(t − ”xy0 l7€‡ |–†xg— tt„ )l¨) ˜ („ a)z €>l  t§ l „ l {†z €‚X w V X†„ ‡ su l l| § s©l x 0)˜ l!x wl”Þ Ÿ | y l€ l x˜ V‡ =€ ß 0ˆ x = 2π a2 sl©™x y ˜ =l 0a†– €(s1l−„ l ˆ€  a 3π ‹ „ ©z € l —†– | ”y l €‚–†V‡ R€ — „ƒ™ xYy 7€ ‡– ˜ l j …x2/3 + y2/3 = a2/3 l “€ — z u ˆ  ÜŠ l©x ˆ3πa2/82.–%—}„l©w©jz„ƒTp„))|‰)V‡0x €€l¨›‘€p–¤y7y7Ÿ0 l©l©lq‡8“³ƒ„(ƒxx|—%Iƒ–ƒ–)´—w88I‡¬†˜w8|e|e©rIc„¤–|)|±)oxY€y˜ŠDxxyoR‡s's8x‡ r(‡|g˜©xe˜d|xDl,c)lel–¤˜yixl›2nnˆ¨¦„)Û a7€t‡‡VO7Ÿ–˜—od€7y˜) —lPw–h‡s|a‡uu|swl—%l¤y=li|‘| d‡‡w‰m|p€§l— orPilrwltƒ„l„5—a„a|>==W x—gryd€ l‡elƒ„0oO0›„8ƒ„sPsdˆ‰| sv–0‡l˜˜’p…–#§Ÿ‡Vllzl–V‡ow„v|„fl 7€lrO‡|| ›!——e„l lXcw„¤|—888zu˜y|}„|w|‰—%€›yr u|l‡E‡‡v˜ wlyx ƒ„a™‡|¤–}„Py‡zs”yw‰x†–—%ˆlO…hl)e‡d0y‡˜…‡Bn€5tX8˜jƒ‡V“³˜2›O€pfl w ›„˜´)ow‡P|†¬l x´–l|†{|ezya…hŠ=8xr‡B„›–˜§‡e)xYx”y)wsθ‡ lx›w%—x !› |e—%P‡l¤|‡| ‰w „ƒl[7y˜)4lx€vl¨0€ll ,l©¤›2l‰˜02xv„‰w B‡l0π–†(l–€””yV‡%—)r€l €‡l,l€7s˜ „eθ}„›„‡l¤y—%)I|–}w‰‡|e©l§fl„}2xO˜‡ „ƒ˜ xlX§‡lxx„ x = r 0 ‡ x θ ; y = xl | θ. rA—%—„}†••tR‹ ‡‡Vw w‡f„¥x€7rg| „ƒ|›!‡Vq–†)0¦…|yˆ€€–u)Œ‡‡y”l©z´}„˜‰–lı|xV‡uVm€”`‡xƒ„€l…–w%—©ll0|e8—%%—¨xR‡V•l„}dzw‡V€2g—‡‡€7˜e€—|g|–¤„ l0„ƒ¤{Izs0˜¢0ƒ„c‡ –ˆθ–€%—‡yz—x¤l•|‡wP›„w‰x”˜g|rlrl[‡€(Dα„›€¤{xzθ–=l‡,z)†–¤0β”€| x‡V€l)w‰r]ll€”x(2lr™€s€θ0lyl©w=}–)‡g|x„ƒsfI||xRD ¨y˜a„聖}wg‡|wθx|w=w”€‡ ˜…¦€”j l…¦avyαl‡2 | j˜s>wre— ‡[θ(8ᄤθ|805=,)|e˜ βŠ,ˆ‡sθ]βθmI—γhs l§E)‡VPx„ˆ©l€ ‡€ lxY©Izr|‰yH„€l θVszlŸ ©„ƒl˜‚€˜›h‡PDxAl |‡g—Ûg0˜5}„‰|˜B‡˜‰‡l+l–V‡ls˜I„€fs „ƒ!γ˜y‘I )l7€l %—–hx0I|)5zd‡‡Vw|w€q”€˜€”ul©|e€ …¦xw›©lxx}„w ©l‡™I— eq|˜%—˜‡s„›l‡–}|}–p‡Ûg|„¤–˜)i¨B‰|7y r˜–€ly–la)l2l˜”€–}|xu„…¦l{†z”yw|8 l˜d†„I|ll ‡‰l|e‡ „ˆ ÷ü}É ¼XÁÅ¿À“Å ËËÊÇÇXBÇ Ë¼ ÌDËÌ0B¼º Ì©ËÇ )À¼$Ë ¼i0Ì Á”ÃSºÀ“н αrÃX¼ güi¼ ú¤ÃË½Ê À&Ç V“ üÁºSÅǽ Ï β 1 r2α 2 A = 1 α ¡ r(θ)k 2 dθ . 2

3.4 APLICACIONES GEOME´TRICAS 119l¿„ gw )zl©€“x l ˜ ˜ xlw |e)——‰7€ —I‡ )¨ €–}y!› ¤– 0 ™ – ‡ z—%| ‡VP€ w=|e{αx w=!› w 0˜ ,lsw)z1€ l,..Rx .˜ ,l wx nl  =y V‡ €β©l }x ˜D–ªl €ˆ%„  †–w | y”l €”©l … x 80}„ ‡‡„› [α‡ ,β] s ƒ„ ©€ A n σ(P) = 1 ¡ r(θj)k 2(wj − wj−1 ) , 2–}|‰˜ | ‡‡ 7y |Dll©˜u x ˆ€l ›) θg¦ zj„ xl ”x ¤lul w‰| ycl j‡l =w›¨d„ 1|e}– !|  y70lx ‡”€w „›¦… !› 8 ‡ „} ‡ x˜ l l ¤ w –l l ¨›ˆˆ€  )|‰l ||4l0V‡„ €”x €w ©l g¦x —%}– | ‡7y |l ˜”€ ¦…– l8|†„ ‡”y l [wj„¤−1• ,w w|e0j–]V‡ z Is| §E21©lrYx2y  ¤ gwσl5(P©l )x ž€– [α, β] ‹„ )z € l  l g|  l 7€ ˆ€  ˜ gs ˜ —l V‡ l7€ –† Ú©w ‡ d | 0 y – ‡z |El©©x)j€ ea7y rl©2cx ˆ ƒ–i8c|eio . j ™%— V‡ €R„¤ mlejmw |Inhiscwa€”t¦… ah d0‡e¢| B•†V‡e€7r›!no u˜ llli ¥ 8†„ Š x2 + y2m 2 = a2 8 j x2 − y2Longitud de arcos de curva„ƒ)Œ q ‚€– 0 „}P‡‡ |(Œ=˜ Iullv)–“{7y lDawgCƒ„B¨ v˜t=yu€ ˜lrw x„›vl”€ 0…a„ƒ‡ ,q xxey„1nj}{z a|=,l.c,h.yay.(r(¤ a,txgwxe))lmns¦‰˜i§a€ˆ=‡n| j˜ bba˜  ,sHl}yx©l (w x5b%— )w ‡m |e| xq˜ l €I— ¦…l ©›2—€ y—‡ –¤x0) –¤‰—‡ z ‰w|7€ ‡l ¨˜ „ƒ}–€l ›!•[wa) €g|,sb0 —e–]‡ z¤„s| ƒ„gw hyx †–}„ P•g¦‡ „| ”l u›(–17y l)wg| j˜[7y alL,s b—%˜‰]‡Vml €„ˆ n7 ¡ y(xj) − y(xj−1)k 2 + (xj − xj−1)2 . σ(P) =t g— „}–}))| ˜ ‡ l „ jj= 1–¥ œ Š ˜ }– • l € l g| 0 ƒ– 8 „ l t| ©  ˜  x w ¦g}– | y”l €”¦… 8}„ ‡ s € ©l x w „ y  n7 σ(P) = ¡ y V (ξj)k 2(xj − xj−1)2 + (xj − xj−1)2 j=1 n7 = 1 + ¡ y V (ξj)k 2 (xj − xj−1) ,˜ ‡| ˜ l j=1 ¤xj‰w)l`ˆ l©™x l l©Yx y ‡ ©l 4x w x”w !›  ˜ l €ž – l ›¨8 ‰| Â| —I)€ˆQ ƒ„ ‰• w | £ƒ„0 q– ‡z •| 7 ξj P €7‡ 7y Dl u ˆ€ ) g¦ „ ‘l l | |e „v†– | ”y l ”€ …8 „}‡ ˆ[a, b] d w‰lDu ‡ x l`y – l | l (xj−1, }– | l d‡z ‚€ › w1„ƒ+ y V (x)k 2 ¡ b7 1 + ¡ y V (x)k 2 dx . L= a„}q‡|†„ ‡ u x –(Hƒ7y …¦Igw ‡8I¨˜|¨„})d‡ ˜€8Cl©l| vx„Iu)z )„}r‰˜ ‡‚€vl uD a%„ ‡— I— e˜© ) n€l€ u 8z„ƒwp›— › a0l rlwy a7€|”€ ‡m…y d t´0e0γ–t‡Vzr§:| icjs¦txa1x (–stl©x)x ,(ty)(tl )ym (lt|)y x ‡l 2| „}‡ • xw |g— Dw –†|‡ y| ‡ )l xx 0” ‡V(€”1€)l©j x[—%t‡0,| t˜ 1– l]m| sy”l)„¤x €  L = t1 7 t0 ¡ x(t)k 2 + ¡ y(t)k 2 dt .

120 CA´ LCULO INTEGRALq}„ ‡ x‘( )IzH |I}„Iu‡ )w | †„ Cu‡ –x y7uwgαr™˜v§ a˜ lβe„Bns) ‚€ p0w ‡ o8|˜lal˜ r‡ƒ„ e€rs w €”…  r j= r(θ) l s l©| x y € l }„ ‡ x — w | y ‡ x DV‡ €”€ l©x —%‡| ˜ –l | y”©l x [α, β]m •”P (1) β7 ¡ r(θ)k 2 + ¡ r V (θ)k 2 dθ . L= α s „ƒ xvl  w 8 D †– ‡ | l)x“ ‰w l©Dx s —I)ˆ€  θP [ α, β] x(θ) = r(θ) D ‡ x θ , y(θ) = xl | θ, r(θ)˜ w(e|IIu)—I8 ) | €ˆy78 l› 7€ –}l0V‡ y€)€7ˆ–})Ú d 0 – ‡z | ˜l „ƒ2 w €”…¦2 h„ƒ2¤ ‰w l — w‰l)˜ l )g— }„ –¤© )€ x l „ƒ„• ‡Vz €7› w „ƒ ˜l„© 8 x| ‡E—I 8)zj¤„€ˆe 8rw ›c}„ ‡il0zcsy i€7%—o–¤©V‡.€ x€ l `§‘Ÿ l l›h}„ V‡4| |g— %—u„}E‡ –‡7y ƒ„wg0©‡‰˜€ |‰l©x‰˜ ˆ¤„2lt w 0 e|†– €‚‘ w D ‰|–ª€‚• l w l‰| |g• lD €¤– l |g0 ƒ–  ˜ l €ˆ ˜ –}‡ Rs$l l©|‰x 2© )π€R7y l©ˆ‘x5ƒ– e8d|e  Dxl $s€ l l | „ € x2 + y2 = R2Volu´menes de revolucio´n(–}ll |a|„ 7y )Vu ll l (I) Por discos ˜ Œ l w l —%l—l ›„€‡wy|‡ ¤–|e©ux ™88 ¤„„„›!€©l lx ‡€¦… x©lx‡ u„ ¤ – =‡wgz0k|l– ‡azty|¢(§ 0x}„‡)–}x„›h› )l =–g—xy „ l0w ˜by|e` ˆ`l—%•|‹‡dw €™€„e|y ‡V0¦…ƒ„ €7`–‡‡Vg|z „|r‡2w w › 08€”‡…„l || y –}| w l| l „ s 8 €… w Ÿ yl ˜ l =„ xy‡z („}l©x– x˜ )‡ Ÿ €”El|¦… ˜8nX„†€ˆ‡  V cX˜ [aa„‡ “sr,t§bew ]s8ƒ„ ˆ`i| ax ˜ƒHn{‡€ ‡al |ts x.y –  ¥ x l XV X b Rx=| „ƒ˜y „ –lπl %—|…¦©‰˜a‡€„l yw¡zsy¤– ›0 —g(– x‡l„ƒz )|| wgk ©l 2xPxR–†dg¦ ˜=x„ll VX . ”y l¦s ¤ w‰hl ˜ –¤ ¥ `‡ …¦‡ „ w › l| ˜ l©l –x l $„ „ {†z ›2y – – 7y l l | s ˜ lw ) | ˜ ¨‡sd˜ „ƒl ˜ 7€ ‡ x= a , x1 , . . . , xn = [a, b]‰|„ƒ“ V‡w‰Rx 7€ l©›¨xx—w !›x ul ˜ ll › l| b} 0 {D x–†}„0}– | n π ¡ y(ξj)k 2 (xj − xj−1) j=1(x}–˜¤ |bwgw‡7y =)|)—ll |˜€”l©E…¦˜lxx 8R‡(n}–t‡|ƒ„j)v”yps[PlDt auy0w [r€ˆ,x€7a)t=…jg¦−m1v „1]l¤y´e,‰s w‰(txl xtrl|j–)il ]„}sc|[†– ˆa›2a0˜ ™,s‡‡– b.y|l R]a€7‡ˆy„ƒ=“ (tl©z t€xYx)yDl• ‡V(uz tD€7x“– 0‡‡›z sx)|| ‡w yt§| xw ¨›  “x ˜ l vž – l ›!8 ‰| 2| —I)ˆ€ 5„ƒc • w g| D– ‡z | s –„¤}|Ew…'¤ – ‰wl |l4§ “l ˜x˜ (t˜)w s‰“w| 0l©—%‡x …¦|‡d‡€ „ ˜wl l› ‚€w –ªld…¦#|0 }–˜‡0 E|‡©l›„Dx ‡‡ %— | ©yl €ˆ–}5„ |8 w8› | π y2 s b = x(t1) 0l”zy yl l 7€€7|¤–}–©V‡ l€ x „ VX t1 2 x(t)dt . = π t0 y(t)k ¡

3.4 APLICACIONES GEOME´TRICAS 121—%l ™„‡ ƒ„ ©8z€}„Œ l© ™–xw „¤„†™2‡ —I˜ w)l ˆ€€”8¦…w ›‘| 0l)lz…hyxY‡7€‰y „¤–zw© ›˜ l l ‰Û0| ‡|‰›„–x ˜†– ›2‡ –†§l„ƒ)|¢€ ¥—%xq‡l „ƒ¤)¥ w‰€d ©l©l  xD˜l s 00˜ ‡‡ ‡ |t| ˜ „ƒr¤–u( θ¥ • )V‡‡‰z €7ˆ˜› l w7€ †–ƒ„…¦h©g¦8 |„ l ”y l 0€7‡}– V‡f| € s0‡I— | yx –†8| | w ˜ – ˜  ˜˜l s ‡ x x ‡z }„ – ˜ ©l(‹‡ xIr|lI|)ƒ„ u‘ Pl |›2o˜r– €ˆx t!› u˜ ‡qbox –ws7y w8 |d ˜0–q‡‡ z w|„¤—¤ € gwlDu!l – ‡lzr|| t l „€˜ 8| y7w l e|7€ }––V‡ 2€… ‰w) lI— „ )y ™€ y 0 ‡˜ h›‡ g— j „ 0l Šy $s l l „“| …¦y‡V‡„ w7€ |‰› ‡hl |8„ ˜l l l „  ŸYV Y b VY = 2π x y(x) dx .¿wg©l xDs yw—gz–¤›0ƒ„ – ‡lwgz ||x –†g¦ „ l › al | y”l¦s˜ –¤ ¥ ‡¨¦… ‡ „ w › l E| ¦… ¨ ˜xll € l „¿„ }{z 2› y – – y7l l| s ˜ lw 8 | ˜ `‡%s ˜‰„ƒl! ƒ„|‰ ‡Vx–7€ !›x w  ˜ lx„ƒ˜ hl %— h… ©‡€„ l©P5x ˜=l {Þx”y 0w %¦=‡ x”aß , x1 , . . . ¨›  , xn = b} [a, b] 0 n π j x2j − x2j−1mRž y(ξj) = n 2π ξj y(ξj) (xj − xj−1) , j=1 j=1˜ ‡| ˜ l s ¤ξw‰j l =l©x x}– j| −y71Dl 2u +€ˆ) x¦gj„ l ˆ l ™l €7‡ l©z Yx y  x„x ‡| x w !›  xh˜ l €ž – l ¨› ) |g|rI— )ˆ€ t „¤¢• w g| 0– ‡z | | ˆ[a, b]2πxyb2E.2j–ej r‹bc„ −i<c¦… ia‡ o„ wŠs.› l 1l .| y ‹ d„ ¦… ‡„ w › l | ˜ l ˜ „ƒ l)x • l 0€  ˜ l l €ˆ ˜ w }– ‡ ‡Rz | ©l x l 4 π†{z R€‚ 3w ˆ }„ ‡ ‡V˜ 7€ lg| „ ‡2to8„rol Ÿ ll | u l | )l €ˆx  ˜ 5‡ —%V‡ €“€ ˆ ¦… ‡ „ 0– ˜ | 3g„  (x − a)2 + y2 −< YV Y 2π2ab2yA2r¨y¦Ig—˜x)´€lπ—I–}lƒ„(„§w!›r8xx)‡Vs)e|eg|Rl©)€u)¦€‚aRx©ytƒHlt¦g0+˜2ys|‡x–„‡l˜˜ll‡—Ilr7€|d2‡|‡s€ˆt|†–) π¦…elDxqlg—€)lj‰sy©ryˆYx„ƒƒ„€7s¤g¦ª{–ª˜zyhŠs‚€€–}u‰w‡wg„Úls¦xl˜¤p„lsxw§2l gƒ„–†Rl‰wƒ„0„}qeg¦‡Vl‡˜‡ll©rx”„„¦Rx†•‡`|l!wxfix‡d|‰y©›|wz˜‚€–πc‡D€˜l˜—!›li‡llrxv|l©lel|!g|xˆ€l‰¥€7sy”y)Ûgˆ™ƒ„ll}–¤€”¥˜d|›„0s€7áÂw‰‡Vy”–}w–‡e‡—%ll™Hlˆ€‡„}–x€ˆc‡d„}˜xlr˜8‡—€r„exI—„€l —g‡˜`)vxl„)z§˜w„ƒ€ƒ„•o€x88lll„l›!yR‡l›7€||‰zR†–u}„–w€™‡s–˜l¤„lc˜|xv˜§|‡I„0i‡7y5lo´y7˜‡}–clll|n„›˜lE€ˆol©w| 7y8Yxl‰n§l)‡yz„l©€”€|eo‡€ ˜„…¦xllsluHl8˜l…hd©l—„†x–}V‡w‡y—ge„›|{„z‡Vƒ„|u„}wl[¤–€“‡rˆ€aˆ€©Dy”cx8e0l–8,o‡•†–yˆ€zv|¤b™l‚€n|€78 ow‰˜]–}„oxÚw5l‡l©lshl‡VuxR„ƒ|”ydy¦‰x)llc–uw˜„€€el|il—%ll|o´t˜˜|re‡ln…¦l€0e˜|o‡‡zvwˆ€€ˆ˜r|‰t„o|ewwe€rz‡˜˜l|ql›mu€xuu‡–}„› ‡n}„)lc˜‰za‡—%‡|€cgil|l‡Vo´xal©du€R€§xnd„ƒe–l–7yS„ƒo)…¦}„˜™gw‡TX|•‡slw˜|€”y„hwg|ulDl)€ˆlwa€‚u—‰–00 7y†–§s‰––˜–l€7V‡wg‡‡‡ezzz©l‡V–}€||˜|‡sxx£s n õ y(xj−1) + y(xj) 7 σ(P) = 2π 2ö ž ¡ y(xj) − y(xj−1)k 2 + (xj − xj−1)2 , j=1

122 CA´ LCULO INTEGRAL§f8„™j 0 ‡¥–| xœ – ˜ Šfl €ˆ—I))€ ˆ€ qw I|„ƒq™•—Iw )|g€ 0y ––¤‡0z |– ‡ z |y(Px)=gs x{l—xjy}–njl=| 0l ˜ l „ –}| y7l €”¦… ) „}‡ ˆ[a,b] t —g}„ ¤– © 8| ˜ ‡ j  c¥ š Š n7 σ(P) = 2π y(ηj) 1 + ¡ y V (ξj)k 2 (xj − xj−1) ,˜ ‡| ˜ l s j=1 l)xYy x 2) w ™' ¨› s — l©z ux ˆ4¡”x© Š“w %— … ‡Vl R€ Ú!ƒ„ )—‰€7‡x¨w }–›!›¨ © ¦g„ ‘l jBliss ˜‰ηlj›„, ξ‡ jxYy P €ˆ ˜ ‡ l |™W[xj−1 , xjô]3 Q ïžj  ñ teorema de 3 n7 σ(P) = 2π y(ξj) 1 + ¡ y V (ξj)k 2 (xj − xj−1) , j=1l©¤ xgw lv©l ¿x w |eˆ  d x‰w w l©¨›u ‡  y7˜ l l lžv2› – l ‡ !› x 8„ƒq‰| „| • V‡zI—7€ ›)ˆ€ w5„ƒƒ„ c • w e| 0 – V‡z | 2π y c 1 + yV 2s ¤ w‰lvl)x –†| ”y Dl u €ˆ)g¦ „ l [a, b] | b7 1 + ¡ y V (x)k 2 dx . SX = 2π y(x) a„ƒu• ‡Vz 7€ƒ › w w 8 ƒ„|  ˜ )l‡hDx sƒ„ h8| w8z „†”€ ‡…¦u  ul l| | l ”yˆ€ ¦l s y 7€ –}Ú ˜ l ¤„  x w — l €”Ûg0– l … l | u  ˜  ˜  l 4| —I) ˆ€ )› l0z y 7€ }– ) ©x s 8› t1 7 x(t)k 2 + ¡ y(t)k 2 dt . =4§á sx l SX = 2π y(t) ¡ x€7lq}– d‡ ˜8€ ˆ l |t—%‡„ƒ© € l©xDs r(θ) sexl —I x „ q—I)ˆ€ 8› Dlz y €7–¤©  Dx s r Ûge|© 8e— „†„†›¤– © lu | ¤„7yulIs• V‡zx t7€– 0› ¤„ „w ƒ„ h w ”€8 …|  ”y lEƒ„  jel „}r†– c— ixcl ioxs2. 1.   xw — l €”Ûe0– „l ˜ l „l }„ –†— x ‡ – ˜ l„˜ l € l …¦‡ „ w 0– ‡z | l | ul | ˜ €ˆ ˜ 2 w 8 | ˜ ‡ a2 j a > bŠ u –†ˆ€  l | y V‡ €7‰| !‡ 8„ l Ÿl s l©x + y2 = 1 b2 XV X S = 2πb b + a2 )‚€  x l | p c , c aq˜ ‡ | ˜l ·   x cw —=l €”Ûga0 –2–l −˜ l b„ 2x ˆ ˜ ‡ ˜l € l …¦‡„ w y 0 – ‡z | l | u l | l ˜ 0€  ˜ ‡%— d‡ € u †– 7€ ‡ ˜ l ƒ„  cardioide a(1 + D ‡ x θ) Xj ˜ ‡| ‡z „}– Š l| V‡ €7‰| ‡!8 „ l %— ‡ ƒ„ )€ 's l)x ˆ32 πa22. ˜l a > 0 Ÿr= 53˜¤ gw‡.xl50‘‡Œ ‡ —l x¦g y–Ixl€ˆN| qy „f„ƒT™ |‰E…¥‡ l‡dGÚ©l€0ˆxY y‰Rj‘˜z lIdAf0… ‡l xLy€cl  E˜„†Rwƒ„ 8S8z!› |l |˜I‡Mll s'|“„ § }–Py| 8y7‡„Rzll©›„€”x …O‡ 8© s‰}„ ‡Pl©x ‡x Ix€I—%s Aw‡‘‡ e|x –†¦gSg¦ – „li™lnD| t˜ IeEl ggۉ| r‰|‡Ra†– €ll©IximEIdf0pM‡ lry t|o A˜p)‡i Nasx ‡Šˆ4‡ ˆxRN„¤l |.x

3.5 INTEGRALES IMPROPIAS DE RIEMANN 123˜xw l  ƒ„l)„˜ ––}‡l| ˜ y7˜lD“u‡ € €ˆ©l )x „™— €wg©l l©x Yx— y lH €ˆy 8‡ ÚD‡˜ e|l $„) g¦†– |„ “ly”l§‰7€ …w‰z8y }„–}‡„'8 ˜ „™l ”Þ }– |‡z y7„› Dl u‡ ß sd˜ ‡lz g¦| –es l l©ˆ€ x–Hg—˜ €l ©l x l s”€ …¦¤ ) w‰€ul ƒ„ R˜ l ˜ –luy ª–x… lD–u˜ w  ˜ | €ˆ8 D – 0†– € ¦ †– € b cb f = f + f.–}w h› | ‰— l€75k|V‡ yg— l d‡ ƒ– ‚€ „›‰|Dx sg‡‡ ˜˜ ˜l‡‰lg¦ „v–¤ l —‰w€ˆ™wl¢| …xy – ‡ s —%‡V€ lŸ ls5›hal —g| „}‡ l s a =x ‡ 5 ˜‰4lc§ 0l ¨ ˜ xYly ›–ª€¨ z ƒ„ x  x!f x ‰|– u‡ w –l©l Yx |y‰y7Qz ©l x8 D –}‡| y”y ©l ˜u €ˆ )„l©l |x l 7€ –†Ûgc© )P € x 5 l – 5„ I© +∞ d1 c d2 +∞ f = f + f + f + f,}–x ›h– l —‰| 7€˜ ƒH‡V‡ −g—‡∞d–ƒu|  1x„}—‡ê< cw −∞ l) ”€ ¤¦gw‰– yl d1 x  ˆ ˜ c d2 ˜ l€˜ ‡ x y –†I— ‡ x$˜ l }– | y”©l u ˆ€ ) „ ©l x 8<„ xdl 2… €ˆI¦)€7 }–xY‡ y –x  w y –†¿€ „ƒ ˜ l Ûg‰| –¤0– ‡z | D7y le”€ ¦… p8„†r¨‡ imd 0e‡ ry a e‡ sp‰˜ led„ c–}i|ey7ê l d f˜ s +∞ f€ˆd0 0‡ – ‡|z |dˆ P 5 § f 8D ‡ y  ˜  l| y‡ ˜ ‡x w g¦ }– | ˜ €”…¦−) ∞„}‡ l }– d| 7y lDu(l˜ a„l ,©l (bxYa7yD]k5,wgs 5lbe˜— ]–}¥‰ls‡ˆ €7le–‡gu ¥ lud‡|0ns‡l d§ˆ€y 8a„f˜ we—%g|s‡lp¤ |˜ w‰e€ yvlc{ƒz‡¨il)e˜ xY„}‡êy}– ›„‡qxawb–¤ y¦g‰wf)–}cl€|0 x‡y7xl l|– €”[¦… a) ,}„ ‡b]880D‡ ‡y y  ˜‡ § l f[ag| ,‡5b8)0‡ ‡ s y  ˜  l | [†– a…¦), b› )l ‡ l|  ˜ ‡˜ € l©x l y €ˆ—‰€)l dzx — | y7l s „¤z `x –}–}|| — ‰w l l l –¤00‡– l s ›l u w‰Rl y – xl ‡| l˜ 7yy7llD€u ©lzˆ€ “x8„y7ld˜ V‡zl7€ –¤‰—0 ‡u€7}– ›¤ y 8 “„ © ˜ l „ y †– —%‡ +∞ f ‰s ˜  ˜ 2‡ ¤ w‰l ê a d +∞ D‡| l „© 8 ›™¦e–†‡ f(x) dx = f(−t) dt x = −t ; −∞ −d b b− 0 ‡ | l „) )q› g¦ –}‡ f(x) dx = f(a + b − t) dt x = a+b−t; a+ a 0‡ | l „©8 ›™¦g}– ‡b− +∞ õ a + bt b − a x−a f(x) dx = f 1 + t ö (1 + t)2 dt t = b−x . a0Definiciones T ‰s x l™˜ w‰l Œ – §f P t [a, T] T T > a S „ ¤{z › f }–  l ¤ ƒ„  integral impropiade primera especie +∞ ©l x T→+∞ a 's §hs %— ‡V€ ˜‰l ‰Û |‰–¤0 – ‡z | s convergente f a +∞ „ {¤z › T f= f. a T →+∞ a

124 CA´ LCULO INTEGRAL t | )z „}‡ u 8› l | 7y Ql Xj xw —I‡ g| – l | ˜ ‰‡  ¥ V‡ ˆ€ f¤ ‰w l „ {¤z › Š svx –f(x) = ∞ f P t [a,t] x→b−§T t P (a, b) S „ ¤{z › t f s¿x ‘l ˜ –¤ l ¤ ‰w l ¤„  integral impropia de segunda t→b− a s'¦§ sespecie b l©x %— d‡ € ˜ l Ûg|‰¤– 0– ‡z | s f convergente a b „ {¤z › t f= f.e§x©ld x0nˆ‡x €7l©y5 †– uVg¦á Š˜wa†–s€“¦s‡| xlD˜–l[|V‡a 7y›2,ul b©ll|‡ x|t]˜—→wlI€l€ˆe|bDz8 5−–„ l¦sxs ˆwwa„†h‡¨›|eRt ¤€ ©xw‰iÛg{zncls ‰|tx—%–eycl g‡V €©lr˜xaˆll lŸ‚€ –}l–ªi|¦„›m7yl lD—gpuf„}rˆ€‡ Po8–s „p)ltx ix– a}„ }–‡Bf„› ©l x(l©‰— 5xc€7‡V)o–†g—sn| –ƒj”yvfl©ex ulro0€0 g©‡cg¦eh| an„… llslmt€elue| lnw| ty7y8e‡©l| ˜x$˜i‡rn˜‡ tl –}xe|‰—l g7y7€ —lr–}›”€agw ¦… bl©l8˜€0l„†e‡l +∞ „ {}z › b „ {}z › c „ ¤{z › b f= f= f+ f,–}0˜ë ›h‡l ‡2|‰Û—‰ƒ”I|‚€{c³d‡l −´e— )∞–¤€”xDu ¦gX{z‹ ê|‰– y„ ‡20€ v© 0a7€ ‡}–lab‡h|o→→s…rl+− €pw ∞∞u )rl |i| na˜y”cl‡ ipxlD™l a¨ l–˜ Yx –¤ayd 8→le|w−dC∞„}i‡ vax euar„ }{czg›„hey–n7y ©ltb˜x2e→l ˆ ˜+Dl∞–„“l €„ƒy c ‡ ˜ l€l  ¥ ‰‡ ˆ ‘ |I4}– | y7Dl u ˆ€ )„ }– | ”y lDu ˆ€ 8 „ ©l x }– h› —‰€7‡Vg— ƒ–  xqx”l ˜ x   ©€0 w |e f P t „}'‡  (5 ) : c¥  +∞ „ ¤{z › T f= f, T →+∞ −T −∞ H  )€ˆ w |e f P t }„ ‡B {[a, c) n (c, b]} : 5¥  c−ε b @b f. f+ „ }{z ›f = a ε→0 a c+ε Œ !l y l `l y”l l €7„–}‡Vx €– l©u xw –0 l ‡| |hy7…l u©l xlwgs „ly y˜ l©x xc– –xwcw e|¥‚‘ lD–}|¨ 7y lDu –€ˆxY8y”„€™l –}§t›hl©—‰x 7€ V‡– ug— w–ƒ8 „$˜ 8 `l „ w…¦)‰| „}‡‡V€ ˜ ˜l x˜ ‡ xvy –†—%‡ x – 8 | | l €€ | ‡‡‰|gê  l „†‡ „ƒ}– | 7y lDu €ˆ8„Xˆ ™l €7‡ l „ l | w g| 0ƒ–  ˜ ‡R€ l  †{z —‰7€ ‡'0 ‡ l©x • 8 „ x ‡ s — ‰w l©x —%V‡ € l Ÿ l h› —g}„ ‡ sÃc¥  +∞ x=0§ c¥  „†‡ u x – § ˆb dx −∞ = −a x j b a m a>0 b>0Ejemplos y ejercicios p > 1 `§ l©Rx ˜ –†… l € ul | 7y lhj  x– x –∞ 11. 1 +∞Š ˆp −< 1 1 xp dx = p − 12. 11 dx = 1 x – p < 1 §‘l©xv˜ ª– … l € ul | 7y l„j  +∞Š x– ˆp −> 1 0 xp 1−p I— )€ˆ ˜‡ ˆ ‡ ∞ ˜ –†… l € ul  +∞ y ‡ p dw‰Dl u 0 1 xp dx

3.5 INTEGRALES IMPROPIAS DE RIEMANN 1253. ∞ 1 x – 's ‘§ ˜ –†… l € u l™x – ˆa −< 0 a e−ax dx = a>0 0 t s „ ‰w l©u ‡ 1 }„ ‡ u x dx = x }„ ‡ u }„ ‡ ux4. ¡ − xk 1 = t − 1 − t t t 1 „}‡ u x dx = „ {¤z › „†‡ u(t − 1 − t t) = −1 . 0 t→0+  © €0 s ˆ∞ ∞ ∞5. n Pr xne−xdx = n xn−1e−xdx ž¢ž¢ž = n! e−xdx = n! k – x  w‰y †– 2€ „¤ l0¨ 0 xY7y l |g0–ƒ ˜‰l „ƒ4 }– | y70Dl u €ˆ8„€–}›h—‰7€ V‡ g— –ƒ ∞ 0 Q —‰€7‡VI¦ ) h€ ¤ w‰l6. – x dx ˆ∞ 0 (1 + x2)p e−² xdx = 2 0 f w 0 ‡|h… l w € ul | y”l |‰‡q–}›h—g„}–¤© q¤ wgl [x0„ →{}z,› ∞∞ )f(ˆ x) = 0 ˆ ‹ g| 0 ‡| y ˆ€ ) € w `| 0 ‡ | y ˆ€  l Ÿ l ›h£ ∞ I| ™• g| 0– ‡z |4|‰„‡ | Dl u  y –†¦…  l|7—g.„}‡20a‡|Criterio de convergencia de Cauchy. Criterios de comparacio´n§ ‹ |E„}‡`¤ w‰ul x – uVgw l s b f x l € z w e| h}– | 7y lDu €ˆ8™„ }– h› —‰€7V‡ —g–ƒ s 0 ‡ | −∞ < a < b −< ∞ f P t „}'‡  [a, b) ˆ aCriterio de Cauchy.  v–}| y7lDu ˆ€ 8 „ b ©l x 0‡|B… l € ul | ”y l ⇔T S t(ε) P [a, b) f ε>0y ) „ ¤ gw lx – s l| y ‡ |g l©x arrrr ˆt2 f rrrr < ε t < t1 < t2 < b ¨ ‚p `o ´8Dn Ò ±0dr ­‚²B´)· q ˆ “ ‰w l©x }„ ƒ„ ) ¨› )| ˜ ‡ t1 sIl ™„ € ©l x w „ y  ˜ ‡ ”x hl ˜ ©l ˜ w  hl ˜ l „¤!0‡| £ ¤– 0– ‡z | j ˜ l €ƒ  ¥h§ Š“—I) ˆ€ u ¤ w‰—l 0l ¨ t „ {¤z › ˜ w  y )'„ ¤ w‰“l x – F– x(yt) =l d„ „ {}za„› f– 7y l Fy (‡ t|g) ê  ˆ€ 8„ ∞ ƒ sdl |t→b− l©vx ƒ ¤ T S δ> 0 „0€7– y70l €7}– ‡ xbl |−δ < t1 < 2nπ ƒ xl |F(t1 ƒ ˆ x t2 < b ƒ ε>0 ƒ„ H}– | y”lDu ƒ˜ †– … l nπ x ) − F(t2) < ε – x •id  ll x˜ l ƒ€d xw  h¥ § ˆ € u ls — w‰©l x V‡ € l Ÿ l „› g— „}‡ s x 1 }– ›hg— }„ –¤©  ‰w l |‰‡ x”lx  y dx−> π  ‡ xvx – uw – l | y7©l x ˆ7€ – y7l 7€ }– ‡ xRx l € l Û l € l |¢ ˆintegrandos de signo constanteŒ l Proposicio´n. § „}‡B)) „}› l | ”y l –}| 7y Dl u €ˆ)¦g„ l–l | ˆ[a, b) b f 0‡h| … l € ul ⇐⇒ f −> 0 S K > 0 y 8„R¤ w‰l F(x) = x ˆ[a, b) j Œ v– „¤¢ –}| y7lDuaˆ€ 8 v„ |‰‡s0 ‡ |h… l € ul s f<KT x P ˜ †– … l € ul  s'§4x™l xw‰l „ l—l)ax ˆ 7€ –†¦g†– € b f = +∞ bˆ Š +∞ a

126 CA´ LCULO INTEGRAL¨ 7p `o ˜ l´80Dn €Ò l0± 0rV–7­l ²'|´©·y7q l ˆ ¿ˆ ¿ á w‰l©l x | y0‡‡ |g|¨ l©ƒ„ ©x s¥ †– — ‡z y”l)x – x“˜ cl xl € s ƒ„ —• w g| 0 – ‡z | ©l x ›„‡ | ‡z y ‡ e| ‰| ‡ f −> 0 F(x) „ {}z › xw —F(x) = F(x) : x P [a, b) ,‘§ l0¨ –xYx →7y lb©l−xY7y l „ }{z 2› – 7y l ⇐⇒ F l©Yx y‰hz d 0 ‡ y  ˜  x w — l 7€ –}‡V7€ › l | y7l ˆ § Œ l 8 | l | ”y l –†—I| ”y)lDˆ€ uˆ€ 8) „g¦ Hu „ ©lzw |cx lc| ö Ë º À©¤úá¼ ‚“ Ç Ë ÅCx‹ w |r—%yi‡‡t|e|grl €“i©l oDx ¤ s w‰dl e S cKom> p0ay r8 a„%c¤ i‰wo´ln0. f, g −>g(0x)}„ B‡ T )x) „}P› P [a, b) ˆ i¼ à ü ”½ ¤À 1‘∞¼ÈÇ D“ X¼ ©À xxú¤¼5d+x 1 −< f(x) K [c, b) [a, b) −< b D ‡|h… l € ul | 7y l =⇒ b 0‡ |B… l € ul | ”y l ˆ g f aa[˜¨ al p7,`obƒ„  )8´ ¥Dnx ҆–‡z—±0„†Vr‡‡z ­7y”²')l˜ ´©·xlq– —x ˆ  V‡˜ l €l„7€ ‡l˜ „ lx ˆl–7€„ƒ`y– y7– ll0|€7‡}–lh|‡ … ˜l l u €ƒl |g Dw ¤– ¥B§¦˜ sl ¤„ h„ƒ 0x ‡–}B|| …7y lDl €u u€ˆl8e|„ l©0x—ƒ–  l ˜ l „“„¤ –}x| ”€y7¦…Dl u) }„ˆ€ ‡8„ l©xRl | l | € | l 7y–}|l ˆ [c, b) tt b f −< K g −< K g T t P [c, b) , cc a„ ‰w lDu ‡ s b < +∞ =⇒ lD¨ – Yx y”l b f s © g— }„ –¤© 8 | ˜ ‡2„¤q—‰7€ V‡ %— ‡ x –}D– ‡z E| 8| y7l 7€ –}‡V)€ ˆ g a c Œ l 8 | s }„ '‡ © 8}„ › l | ”y l –}| ”y l £ ö Ë º À©úƒh¼ ‚“ Ç Ë Å lˆ i¼ à ü ½”¤À 1‘∞¼ÈÇ “DX¼ ©À xxú¤¼5d−x 1Cu €ˆr) ig¦ t„el©rxviol | de cbo)mQ x pw —%ar‡ a| cl €5io´¤ n‰w l po„ r{¤z › l´ımite. = f −> 0 g > 0 [a, f(x) x→b− g(x) l € ul Œ s l ul y7l € u l | | ”y l ¢Vˆ – b f 0‡|h… l € ul | ”y l ⇐⇒ b g 0 ‡h| … y7l ˆ ˆ  Bˆ l>0 a 0 ‡ B| … € | =⇒ a 0‡h| … l Œ s – b g b f l=0 a aTE}–x¨ |l—xjy7p7eylD`o1∞–rPu l ´8cˆ€ | [Dn)1iclcÒe¦g,+±0i1„gboVrf—l/(7­x(s)xxx²'x.2sg´©–)·)q§kd§¨−<„ˆx–)x‰x C;‡Vƒg—z „}w}„w‡–¤Ty1©∞8 †–x8x|€5–| ˜ ƒ„ƒgP˜xh‡ lx‡ [l©|02cS x‡l,xx„$h|}–bƒ→|„…8¤{)zdy”›l|bl©ˆx€7y−uul;0€ l€7gf©|g}– ((g¦‡V1∞x0x—€„ ƒ–)l) ˆHˆ1t€7>˜ ƒ– 0y7l w‡l sdxl ˜ ‡hl¦… d¨ 0¤ ‡ w‰ h› l 7yI„—Il „|)}{z ›„€ˆl d€ –0”y 0–l™‡z |<©l x jC˜ l1‡ 7€ x ‡−<…s f(x) s −<l f| C7yl©2lx 0€7)}– |‡ ˜ glx ‡(l©x„¤8x)›Š ƒ„  Rx x – uVw – l | ”y l)x –}| y”©l u €ˆ)„ ©l x }– h› ‰— €7‡Ve— ¤–  x ê x‚õ 1 2 dx ∞ dx dt, ; ; †„ | }„ |t ö 1 x 1 (1 + x) ; }„ |∞ dx 1 dx ∞ ƒƒ ∞ x)p ·; 2 ; cosx xp−1 e−x dx . 0 (1 + x) 1 − x2 ·; x( 0 1 − x2 0

3.5 INTEGRALES IMPROPIAS DE RIEMANN 127Las funciones Gamma y Beta de EulerD§ ue¤fiwgnl icx i–l o´yn– .l |  l v–†| ”y lDu ˆ€ 8 B„ }– „› ‰— €7‡Vg— –ƒ +∞ 0 ‡ |B… l € u “l x – §ux ‡z „†‡ x– s ê xp−1 e−x dx p>0 0  „ }{z › xp−1e−x = 1, § „ {¤z › xp−1 e−x = 0 T s. x‰w →l0˜ +l gÛ | xlps −I— 1)€ˆ pP (0, +∞)xs→ƒ„ +q∞• w 2‡ ¤ g| 0– ‡xz |s +∞ Γ (p) = xp−1e−xdx , ¤ ‰w ™l xl D ‡V|gB‡  l 00 ‡„› ‡ ˆfuncio´n Gamma de Euler¢VˆPropiedades. ˆp > 0 jš | y7lDu ˆ€ ) v€ —IV‡ €v—I)€ y”l)Dx s xp = u ˆbŠ Γ (p + 1) = pΓ (p) T  Bˆ Γ(n + 1) = n! I— © 0€  n = 0, 1, . . . ˆ Γ ™P ” (s ∞—I))(€ˆ5  + )‡ s$Q§ x !l y – l| l Γ (n)(p) = „}‡ u∞ x)ndx —I) €ˆ4 ©  ˜  n ˜ ‡ p> xp−1e−x( Pr y 0 ˆ 0 ˆ ê „ {¤z › np n! . Fo´rmula de Gauss Γ (p) =y‘ ‡Vz• €7w › w w „‰Û y g|–†g— – ˜}„ }– ) %—y –†V‡ …R€ –‡ —Iy )€0€ˆ™ 5–}|„ƒcy7Dl • uw €ˆ|gn8 0→„– ‡–†∞z ›„`| —‰p–~ €7(V8‡p›„g— +ƒ– ›¨Bˆ1 )Œ ¥ž¢l )¢ž8ž€| (–z p)+—I©n€ l) 0l f€ ê ‡ y ) e|| 5”y l • g| 0 – ƒ„‡zc| › ˜ l €ˆ5}– ›h—%V‡ €”£ p, q > õ +∞ õ +∞ Γ (p)Γ (q) = xp−1e−xdx yq−1e−ydy 0 ö0 ö = +∞ õ +∞ xp−1 yq−1e−(x+y)dy dx ö 00 x+y=u = +∞ õ +∞ dx xp−1 (u − x)q−1e−udu ö 0x Fubini = +∞ õ u e−u xp−1(u − x)q−1dx du 0@ H0 ö +∞ 1 x = ut = e−u du up−1tp−1(1 − t)q−1uq−1 u dt 00 @ H õ +∞ 1 = up+q−1e−udu tp−1(1 − t)q−1dt Hö 0 0@ 1 = Γ (p + q) tp−1(1 − t)q−1dt . 0

128 CA´ LCULO INTEGRALDefinicio´n.  „ –}| y7lDu ˆ€ 8$„ –†„› —‰7€ V‡ g— ƒ–  1 0 ‡|h… l € uul x – §¢x ‡z }„ ‡ x– xp−1(1 − x)q−1dx § q > 0 § u ¤ gw lxl–y – l | lê 0p>0 „ ¤{z › §xp−1(1 − x)q−1 „ ¤{z › xp−1(1 − x)q−1 xp−1 = 1, x→1− (1 − x)q−1 = 1. x→0+ 2‡ ¤ ‰w l˜ l Ûg| l s I— )ˆ€  p,q P 5 + s ƒ„ q • w g| 0– ‡z | 1 B(p, q) = xp−1(1 − x)q−1dx ,¤ gw l x l D ‡ |‰B‡  l 0 ˆfuncio´n Beta de Euler 0‡„› ‡Propiedades. B(q, p) j )) q› g¦ }– ‡ y = 1 − x ‚Š ˆ ¢Vˆ B(p, q) = B  ˆ Ž y ˆ€  x ƒ• V‡ €7›¨ Rx ˜ l ƒ„ u• w |g0– ‡ z | ٓl0y  j ©8 ›™¦g}– ‡ x x = 0 ‡ x s2 θ x = y/(1 + y) ˆŠ ê 0 ‡ x x l |π 2q−1 θ dθ 2 B(p, q) = 2 2p−1 θ 0 = ∞ xp−1 = 1 xp−1 + xq−1 0 (1 + x)p+q dx 0 (1 + x)p+q dx . ˆ j§ B(p, q) = Γ(p)Γ(q) ¥ q¤ w‰©l ˜ ˜‡ ˜ l ›„‡ Yx y €0 ˜  Šˆˆ Γ (p + q) ˆ Γ j 1 m · πQ ˆ∞ e−x2dx = · π 2 = −∞ ¿ ‰w l©x —%‡Vu€ ƒ„ ¢8 | y”l €7†– V‡ € s pΓj 1 2 j 1 1 ˆπ/2  ‘‡ y ˆ€  †– | y”lDu €ˆ8„ ©l x  Dx {z ê 2 2 2 m q = B , m = 2 dθ = π 0 ∞ e−x2 dx ∞ õ 1 =2 e−x2 dx = Γ 2ö , 0 −‡|∞l „©8q› ¦g–}‡ 0 x– ˆx2 = t B© ˆ Γ(p)Γ(1 − p) = x l | π ˆ0 < p < 1 (pπ) 'ä ˆ êFo´rmula de duplicacio´n para la funcio´n Gamma 22p−1 õ1 Γ (2p) = · Γ (p)Γ p + . π 2ö

3.5 INTEGRALES IMPROPIAS DE RIEMANN 129¿ w‰)l x s“§hsj 2 0‡ #| „ƒf•†V‡ €7›! y 7€ – u ‡|‰‡ › 0lz y 7€ –¤© ˜l ƒ„ f• w g| 0– ‡z |“Ù 0l y  s Γ(2p) = Γ (p)m B(p,p) xl | x l |π/2 πB(p, p) = 22−2p ( 2θ)2p−1dθ = 21−2p ( t)2p−1dt x l |0 0 π/2 = 22−2p ( t)2p−1dt 0 Γ j 1 m Γ (p) Γ 2 = 21−2pB õ 1 , p = 21−2p j . 2ö p + 1 m 2Ejercicios. 5 + Û ŸY‡ sV¢ ˆ  ©€ˆ ν P ˆ∞ xνe−pxdx = 0 Γ (ν + 1) T p>0 pν+1B  ˆ 1 õ }„ ‡ u 1 n dx = n! I— ) €ˆ n Pr ˆ 0 xö ˆ5€ƒ 8„} w „ƒ© 5€ ƒ„  x –†| y”lDu ˆ€ 8„ ©l x In = ∞ tne−t2/2 dt —I)€ˆ n Pr ˆ −∞ˆ ˆ∞ e−xn dxB© dˆce ­7) 0X² }„f҃„ ç)ˆŠ € ˆ l „ )z € l ` }– | ”y l 7€ }– d‡ „€ 8„ ‡dz …¦8 „}‡ jy2(1 + x2) = 1 − x2 ¤¼ ñU2Ç1 Í 0ó 3 ï “s ´ˆ³ ç ‘ |e € l)x — ‰w ©l xYy  l©x ˆ1ä'dˆce ˜) xDl}„ }{„ƒz |)l©€yxY‡y yul „Q w )¥z ”€€8¦…l †„„ƒ) l€l |l jj | „ey€B‡V¦…l 7€‡ ¤„|‰„41w‡!,›c21ilsm”x|ow−i˜d Blex©„ {}z | 3 , 2 m y 4 j 0l Š §`u –†x€7w‡ ™‡ a—%>V‡ € dx ‡yze‡„}– y ˜DB‡ ˆ io˜ cl l€el s¦… y‡ „2w (0a– ‡z−| xu )l |=l €ˆx 3˜ „¢Dà¡ã'BÜ dcˆˆˆˆ e y0l˜)¨„}xDl2‡‡—E„ƒ{}z—g()|| ›2}–1€€uy| ©lw‡+––‰2xy7x€”y –l0gw„›¦…‡lxz¨˜4x—„‘|)q‡– ‡V)jXz˜yz=gÛ¤|€ l3|el‰w €7128˜vl„ƒ‡V−„−—l l}„ „}}–l‰wlx|–}›„ex4|„›©l3mlYx–ˆ–•y„ey=yw‡hn88g|˜ ||i—%10s y7–‡dclw‡ zl§€a€7|I||}–tV‡Y`y¤„a˜€€ ¹gllshx€ d„%©lDÎ „ƒyue‡…¦¨)ǖ|„83‡|Bz}„|‚ñƒ„‡VuVw e€l'U”€ rd|v…'Γn0y7edž‡©loˆŠjryx(ˆ31usaim¢˜le2lˆ hirƒ„−sˆa¢c(exx8d›„„†22e„ƒ–)) +xy€M!› 2y a2=l)l G„2|x)=4za€xkelgw at2xja2s“k(n˜ ˜x−>a‡ l2xŸ”1A−8Š—| gwy§s˜ n|2‡Exey)wg‡sˆ¤„ xxi¢V¢ ˆ5€ƒ 8 „} w ƒ„ © € ∞ j €ˆ18−X„ ˆ e−t m stq−1dt —I©0€ ™}„ ‡ x …¦8„}‡d€ l©Rx ˜ l q¤ w‰l—¥ d  l 4| 0‡|h… l €”£ ul | y7l „¤q }– | 0”y Dl u

130 CA´ LCULO INTEGRALIntegrandos de signo variableDefiniciones. Œ l  f P t „}‡B [a, b) Q x”l¨˜ –¤ l ¤ ‰w l „ƒ` }– | y7lDu ˆ€ 8 „H†– ›„‰— €7V‡ g— –ƒ b l©x fabsolutamente convergente  w 8| ˜ „‡ „¤ b ƒ ƒh©l x D ‡V|B… l € ul | ”y l ˆ a f ƒ w 8 | ˜ ‡ b fcl©ox n0v‡e|hr…gl e€ nu tl |ey”ˆ ls — l €‚‡ b ƒ a es x l™˜ –¤ l ¤ ‰w l b l©x f con- a ƒ = +∞ adicionalmente f aProposicio´n. b ) ¦ x ‡„ w y 8 › l | y7l 0‡ |B… l € ul | ”y l =⇒ b l©x D‡ h| … l € u l | 7y l ˆ f p7o`´8nDÒ 0± Vr 7­ '² ´©· q f y”l ˜l w ¥B¦§ s sal ‡l g|5„  ˆ )l €7x– [a, b) y 8R„ ¤ ‰w l4x –¨ < t1 < t2 < ˆ V‡ | € y €‚†– ‡ ƒ€  T ε > 0 a Pt b S t(ε) rrrr t2 ƒƒ rrrr t2 ƒ ƒ f = f < ε. ™ l €7‡ l | yt1 V‡ e|  l©xDs t1 0 ‡ h| … l € u lsrrrrt2 f rrrr −<t2 ƒ ƒ b <ε =⇒ f f )g— „}¤– ©8| ˜t1‡„‡ y ˆ€ ™t1… l Ú l „ ˆ7€ – y”l €7}– ‡ ‰˜ l €ƒ aw  B¥ § ˆ 0‡ x x 0‡ xrrr  € –†| ”y ©l u €0) „ ∞ y”lxˆ2 l©x 0 ‡h| … l € ul | y7l s — gw l©x 0‡›„‡ x rrr < l | 7y l 0‡|h… x2Ejemplos. 1.– u1l | dx1 s l©x ) ¦ x ‡ „ w y ) › l €x2   }– | xYy y7lDu €ˆ8„ ∞ x l | x l©x 0 ‡ | ˜ –¤0–}‡ |e8}„ › l | 7y l 0‡h| … l € ul | y”l Q 8 | y”l 7€ }– V‡ €7› l | y7l Rxl …'| – q‡ ¤ w‰l ¤„ – 0}– | 7y Dl u x€ˆ8„ d˜ lx„ … 8„}‡V€€ )¦ x ‡„ w y ‡ ˜ –ª… l € ul Q —%V‡ €f‡ y 7€ ‡™¤„  ˜‡s „ƒ–}– | y7lDu €ˆ8„2xRl .–¥ x ‰| ‡ l©x –}›h—g7€ V‡ —gƒ–  Q l –}| y7Dl u ˆ€ 8| ˜ h‡ %— V‡ €vI— ) € y”©l xvl d„ ‡ y 7€ ‡ xw ›!8| ˜‡s dx 1 0 x xl | 0‡ x 0‡ x x ∞ x ¢ − x£ ∞ ∞ x2 1 dx = − dx ; l „› ‡ 1˜ ‡2¤ xw‰l ƒ„ q}– | €ˆ8„ 0 x‡h| … l 1€ ul ˆ˜ ”y Dl u0‡B| … lt € u 0 l ‡e| | 0y ƒ–}–| w˜ dl 0 –}– |‡z 7y|4Dl u… ˆ€l 8›„„ )l ‡ x x 0 „}‡‡ |4x5˜}– | ‡ ”y xlDu0ˆ€€78– 7y | l ˜€7}–‡ ‡ x˜ l› x z Rxu |‰gw ‡„x w … 8 ) „€7)l –ƒx) ¦gI— „)l ˆ€ ê  l„ l©xYy”we˜ }– ‡ ˜ l „ƒ –ˆ Œ lCriterio d‘e—%D‡Vh€ ir‡ iyc€ˆh‘ leI— t) .€ Œl gf›„P ‡t| z }„ y'‡ ‡ |e[a , lb| ) s …l 7€ †– gÛ ©8 | ˜ ‡2¤ w‰l „ {¤z › ˆrrrr c f rrrr −< K T c P y7l §Ay 8 v„ ¤ gw l a[a, b) ‡ [a, g(x) = 0 b) x→b−‹ | y ‡ g|  l©Dx s ¤„ u–}| 7y lDu €ˆ8„ b ©l x 0‡|h… l € ul | y7l ˆ fg a

3.5 INTEGRALES IMPROPIAS DE RIEMANN 131 ¦…¨ „©¤{‚pz€›`oƒ{z f ´8gnDEÒ(0±x©dr ))­‚¦%B²=´)‡f· q 0„ƒˆ  ‰s Œx˜—lw l %— ˜2› ‡l)‡|˜ xuw y 8€ˆ„›ld0 ‡¤– ‡xgwz #|l—%‡V¤ € ‰w l4l Ÿ lx „›–l u| w‰—g[l„}a‡ s,D b‡¤ |w)‰w l§ „ƒ gx ‰w l)•lw–x |gT ˜ 0εl }– ˆ‡>€| l ©l 0x – l | y”`l §jx ¿– |‰‡ Š dsyQ )x 0l„ ‡¤ }„„› „w‰l ‡l £ g ¤ −> 0 0 S −g −fx→b− ˆ c P [a, b)0 −< ggwŒ(xll 8)| ε T xy ©l(cx ,¤ b‰w )l §< Py )„ ¤ 2K t2 c < t1 < t2 <bQ )—g}„ ¤– ©8| ˜ ‡ j ¥–œ   Š s S s P [t1, t2] t1 8„ t2 s fg = g(t1) f ; „ ‰w lDu ‡ t1 t1 rrrr t2 fg rrrr = g(t1) ž rrrr s f rrrr −< ε õ rrrr s f rrrr + rrrr ˜ l €ƒ t1frrrr 2Kε = ε, 2K <§ ƒ„ u }– | t”y 1©l u 0€ ) „d… l 7€ –†Ûg©  l „ˆ€7t– 17y l 7€ }– ‡ ˜ l 0 ‡h| … al € u l g| 0 –ƒ a ö 2K w  h¥ § ˆŒ –Ejemplo.— g l©x„w |I¢• w g| D– ‡z ‚| „› ‡ | ‡z y ‡|e l| § „ {}z › s g(x) = ƒ„  [0, ∞) 0 –}| y7Dl u ˆ€ 8 „ ∞ xl l©x l € u l y7l x→∞ g(x) | 0‡ h| … | ˆ x dx 0CŒ l rtit%—eV‡r€2io‡ dy €0e4 A—%©b€ ey7ll. Œ l„›  ‡f| ‡Pz y t }„ B‡  § [ad,0b‡ )y s … l 7€ †– Ûe© 8 | ˜ ‡¨¤ ‰w l | aby ‡ fg| ©l x)l xD0s ‡„ƒ|hE … l–}|€ u l | y7l ˆ ‡|e  ˜  l | ˆ[a, b) ‹ ”y l©u €ˆ)„ g b ©l x 0 ‡|h… l € ul | 7y l fg ˆ ¨ a‚p `o ´8nDÒ 0± dr ‚­ B² ´)· q ˆ  V‡ € x l € g ›„‡| ‡z y ‡|e § 8 D‡ y  ˜  l | sBDl ¨ – Yx ”y l–l „„ {¤z › g(x) = x w — ‹g(| xy )‡|g=l©Dx λs ˆ |g0– ©l x x→b− ƒ„ t• ‡z | ϕ(x) = g(x) − λ [a, b) w „› ‡ | ‡z y ‡|e § „ {¤z › s „› – l | y €ˆ x ¤ wgls —%V‡ R€ ‡ y 7€ ‡„¤„  ˜ ‡ s b f 0 ‡ h| … l € u l | 7y l }– ›hg— }„ –¤© x→b− ϕ(x) = 0 a rrrr c f rrrr −< rrrrr b f rrrrr =K TcP [a, b) ; a a ˜ ‡7y l ls‰„gxˆ—l €7–y 7y– ll €7| –}l‡ ˜ Rl k –†€7¤–  ¥ „ l0)y s € l©x”w „ y –¤ wgl )l x 7y Dl u ˆ€ ) g¦ „ vl l l ‚€ ‡ sÛg)g—|e}„)–}„})› )| l | –}| | ˆ[a, b) ϕf bb b gf = ϕf + λ f . aa a

132 CA´ LCULO INTEGRAL La integral ∞ x l | x dx. Lema de Riemann-Lebesgue áf x ) ¦ l ›„0‡ x s ¤ x‰w fl ©l xYy ‹Q| –}y| y7Dl u )l€ˆ8Dx s„ )l x 0 ‡ | ˜ –¤0–}‡ e| 8 „}› l | ”y l 0‡|h… l € ul | y7l ˆ Œ l ) | ˆP r ‡ |g tn = (2n + 1) π n 2 xl |π ∞ xl | „ ¤{z › x l | „ ¤{z ›tn p j n + 1 m z 2 x x dx x=(n+ 1 )z q dz dx = = 2 0x n→∞ 0 x n→∞ 0 z xl j @ H  |π  „ {}z › xl | x”l x l |= p n + 1 m z − 1j 1 p j  π |n→∞ 2 2 − 1  2 2 j q n + 2 m z dz = , z z z 0 2 m 2 m q — w‰l)x —%‡V€ w 4| „¤ ˜ ‡ s %— © ˆ€  y‡ ˜ ‡ xl … l 7€ –†Ûg© n xl |π p j n + 1 m z xl | 0 ‡ x 0 ‡ x0 2 2 j q dz = πõ 1+ z + ž¢ž¤ž + (nz) dz = π 1 dz = π , z 2 ö 02 2 2 m 0§ —%‡V$€ ‡ y €7‡ s „ƒR • w |g0 – ‡z | –}ϕ| w(z )s =„ ‰w Dl 2u x‡„l |}–1| j7y lD2zu €ˆ−)g¦ 1z„ l0ls ¨B`§ y7ll | ˜ – ˜ 5 [0, π] › l©˜ –ƒ8| y”l ϕ(0) = s l©x D‡| ‡ |g l)x „ {}z › ϕ(z) y m | y =0z→0 @ H „ ¤{z › π xl 1j −1 xl | j –l | z p n + 1 m z dz = 0 , n→∞ 0 | 7y l 2 2 z m ›¨hê q )g— }„ ¤– © 8 | ˜ ‡ l „ x 2 l – u w „L„ƒu e•mw e| a0– d‡Vz |e ϕRe|‡ iP˜‰eϕlmt –†|a[D a”ynV‡ ©l,|nbu y –€0]}– |8Lˆ 0w e– ‡ bz s4| e„}`‡s%— gV‡ ‰— uv€ €‚e‡dI— .¦) ‡€ Œ hl y – l | l „ {¤z › b xl | (ωx) dx = 0 sx –w |‘€ l) x)w ˆ€„ y  7y ©lk x †–ê€7¤–y  ϕ(x) sl©x ω→∞ a ” (1) j [a, b]m w˜ z ‡V¥ ¨› „ 0l 8y |ˆ ˜  ‡ ) ˆ€  w |e ϕ s P u = ϕ(x) b = 1 b 1 b ω a ω xl | 0‡ x 0‡ xϕ(x) ϕ V (x)— l 7€ ‡ a (ωx) dx ¡ −ϕ(x) (ωx)k + a (ωx) dx , D‡ xrrrrr (ωx) dx rrrrr −< bƒ ƒ b dx = A, ϕ V (x) ϕ V (x) a a ˜ –u Dx s § w‰l ƒ ƒh©l x y –†| w s ©l x ”y Dl u €ˆ)g¦ „ l ˆ ‹ y ‡|g )l xvx –l y – l | l 8„› ‡ u¤ D ‡ „› ‡ 0 ‡ |  –†| | ϕV „ ¤{z › 0‡ x1 b (ωx) dx = 0 , ϕ V (x) ω→∞ ω a

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