HOMEWORK 6 - INTEGRATION. READING: Read the following parts from the Calculus Biographies that I have given (online supplement of our textbook):

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1 MAT 3 CALCULUS I 5.. Dokuz Eylül Uiversiy Fculy of Sciece Deprme of Mhemics Isrucors: Egi Mermu d Cell Cem Srıoğlu HOMEWORK 6 - INTEGRATION web: hp://kisi.deu.edu.r/egi.mermu/ Tebook: Uiversiy Clculus, Erly Trscedels, Joel Hss, Murice D. Weir, d George B. Thoms, Jr., Ieriol Ediio, d ediio, Perso,. READING: Red he followig prs from he Clculus Biogrphies h I hve give olie suppleme of our ebook):. Hisory of he Iegrl. The Fudmel Theorem of Clculus 3. Hisory of Differeil Equios 4. Biogrphies of he followig mhemicis d scieiss): Crl Friedrich Guss ) Georg Friedrich Berhrd Riem ) George Dvid Birkhoff ) Richrd Dedekid 83-96) Afer you hve sudied our ebook, you my i ddiio use he followig issues of Memik Düysı which coi some subjecs of iegrio h we shll sudy his erm: ) MD -II, pges b) MD -III, pges c) MD -IV, pges 4 3, d) MD -I, pges 3. e) MD -II, pges f) MD -III, pges 7 6. SOLVE THE BELOW EXERCISES.. Use fiie pproimios o esime he re uder he grph of he fucio f) = 4 bewee = d =, usig ) lower sum wih four recgles of equl widh b) upper sum wih four recgles of equl widh. Evlue he followig fiie sums ) k= k b) k= k3 d) e) k= kk + 3k ) k= k3k + 5) f) k= + ) k= k c) 3. Fid he orm of he priio P = {,.6,.5,,.8, }. 4. Epress he limi lim k= P c k 3c k) k, where P is priio of [ 7, 5]. 5. Iscribe regulr -sided polygo iside circle of rdius r d compue he re of oe of he cogrue rigles. Compue he re A of he iscribed polygo. Compue he limi lim A, d compre he re of disk of rdius r. 6. Le f) be coiuous fucio. Epress lim [f )+f )+f 3 )+ +f )] s defiie iegrl. 7. Evlue lim by showig h he limi is Evlue he followig limis:

2 ) lim b) lim. 4. c) lim si π + si π + si 3π + + si π ). d) lim 9. Grph he iegrds d use res o evlue he followig iegrls: ) ) b) ) + ) + 3 ) + )). 4 c) + 4 ) d) + 4 ). Wh vlues of d b mimize he vlue of b )? Hi: Where is he iegrd posiive?). Wh vlues of d b miimize he vlue of b 4 )?. Use he M-Mi iequliy o fid upper d lower bouds for he vlue of 3. Use he M-Mi iequliy o show h if f is iegrble he ) f) o [, b] = b f) b) +. f) o [, b] = b f) 4. Use he iequliy si, which holds for, o fid upper boud for he vlue of si. 5. If vf) relly is ypicl vlue of he iegrble fucio f) o [, b], he he cos fucio vf) should hve he sme iegrl over [, b] s f. Does i? Th is, does b vf) = b f)? Give resos for your swer. 6. I would be ice if verge vlues of iegrble fucios obeyed he followig rules o iervl [, b]. ) vf + g) = vf) + vg) b) vkf) = k vf), where k is rel cos. c) vf) vg) if f) g) o [, b] Do hese rules ever hold? Give resos for your swer. 7. Le f be fucio h is differeible o [, b]. We hve defied he verge re of chge of f o [, b] o be fb) f) b, d he iseous re of chge of f ech [, b] o be f ). O he oher hd, for fucio g h is coiuous o [, b], we hve defied he verge vlue vg) of g o he iervl [, b] o be vg) = b b g). To mke cosise hese wo coceps, i will be resoble o hve fb) f) b = vf ), h is o hve, he verge re of chge of f o [, b] equls he verge vlue of f o [, b]. Is his he cse? Give resos for your swer. 8. Aoher moivio for he defiiio of he verge vlue of coiuous fucio y = f) o iervl [, b] is s follows. Le Z +. Mke priio of [, b] by dividig i io iervls of legh b )/, h is, ke he priio P = {,,,...,, } where k = + k b for k =,,,...,. The k = k k = b for ll k =,,...,, d so he orm of he priio P is P = m{,,..., }) = b )/. Choose poi c k from ech iervl [ k, k ] for k =,,,...,. The verge vlue of he vlues fc ), fc ),..., fc ) is fc ) + fc ) + + fc ). Wh is he limi of his s? Hi: Is his like Riem sum?) 9. Is i rue h every fucio y = f) h is differeible o [, b] is iself he derivive of some fucio o [, b]? Give resos for your swer.. Use he formul m k= si k = cos h cosm+ )) si o fid he re uder he curve y = si from = o = π/ i wo seps: ) Priio he iervl [, π/] io sub-iervls of equl legh d clcule he correspodig upper sum U; he b) Fid he limi of U s d = b )/.. Evlue he followig iegrls:

3 ) + ) b) c) π/4 d) π/6 sec + ) e) 3 y 5 y y 3 dy f) 4 4 g) π cos + cos h) l e 3 i) / 3 +4 j) 4 π k) π l) 5 + m) π/3 si cos. Prove he Leibiz s Rule: if f is coiuous o [, b] d if u) d v) re boh differeible fucios of whose vlues lie i [, b], he 3. Fid he derivive dy ) y = si d b) y = sec d c) y = 3 e d d v) u) of he followig fucios: f) d = fv)) dv fu))du. d) y = 4 e) y = e f) y = si cos si cos d d d g) y = si d h) y = +4 d 3 +4 d 4. Use Leibiz s Rule o fid he vlue of h mimizes he vlue of he iegrl +3 5 ) d. 5. Fid f) if f) d = Fid f4) if f) d = cos π. 7. Suppose h f hs posiive derivive for ll vlues of d h f) =. Which of he followig semes mus be rue of he fucio g) = f) d? Give resos for your swer. ) g is differeible fucio of. b) g is coiuous fucio of. c) g hs locl miimum =. d) g hs locl mimum =. 8. Use subsiuio formul o evlue he followig idefiie iegrls: ) + 5) 4 b) + ) /3 c) sec d d) cos ) si d e) 9r r 3 dr f) cos g) 4 3 h) 5 i) 4 j) 3 + k) + 5) 5) /3 l) cos θ θ si θ dθ m) si θe si θ dθ ) l 9. Use subsiuio formul o vlue he followig idefiie iegrls: d e) The grph of g hs horizol ge =. f) The grph of g hs iflecio poi =. g) The grph of dg/ crosses he -is =. o) dz +e z p) 4 q) 5 9+4r dr r) e rcsi s) 8 sec + 3 ) ) e θ dθ ) π/4 sec b) c) π/ 5r r +4) dr si w 3+ cos w) dw d) π/6 cos 3 θ si θ dθ e) 3π/ π co 5 θ 6 sec θ 6 dθ f) π/4 + e θ ) sec θ dθ g) π/ h) π/ cos θ π/ +cos θ dθ i) l 3 e +e 3. Skech he followig regios d fid heir res: j) k) 3 /4 l) / 4 4 s ds ds 9 4s dy y 4y m) sec sec ) ) he regio bewee he curve y = 3 3 d he -is,. 3

4 b) he regio bewee he curve y = /3 d he -is, 8. c) he regio bouded below by he curve y = + cos bove by he lie y =, π. d) he regio bouded below by he lie y = / d bove by he curve y = si, π/6 5π/6. e) he regio bewee he curve + y = d he -is i he firs qudr. f) he regio eclosed by he curves = y d = y 3. g) he regio eclosed by he curves = y y 3 ) d = y y. h) he regio bouded by he curve y =, he lie + y = d he -is. i) he regio eclosed by he curves y = d 5y = + 6. i) he regio eclosed by he curves y = 4 d y = / + 4. j) he regio eclosed by he curves y /3 = d + y 4 =. k) he regio bewee he curves y = si d y = si, π l) he eclosed by he curve y = si π d he lie y =. m) he propeller-shped regio eclosed by he curves = y /3 d = y /5. ) he regio i he firs qudr bouded o he lef by he y-is, below by he curve = 4y, bove lef by he curve = y ), d bove righ by he lie + y = Suppose F ) is iderivive of f) = si, >. Epress 3 si i erms of F. 3. Show h if f is coiuous, he f) = f ). 33. Le f be coiuous fucio o symmeric domi [, ]. Show h: ) f) = f) if f is eve b) f) = if f is odd. 34. Suppose h f) = 3. Fid f) if ) f is odd, b) f is eve. 35. Le f be coiuous fucio. Fid he vlue of he iegrl I = f) f)+f ) by mkig he subsiuio u = d ddig he resulig iegrl o I. 36. Prove h, if f is coiuous o [,b], he b f) = b c c f + c) for y c R. This propery ells us h, defiie iegrl is ivri uder rslio. 37. Fid he curve y = f) i he y-ple h psses hrough he poi 9, 4) d whose slope ech poi is Fid curve y = f) pssig hrough he poi, ) d hs horizol ge lie his poi, d y = 6 for ech. How my curves like his re here? How do you kow? 39. Epress he soluio of he iiil vlue problem y = sec, y) = 3 i erms of iegrl. 4. Solve he followig iiil vlue problems: ) y = 4 + 8) /3, y) =. b) y = 3 cos π 4 θ), y) = π Show h y = + sec ) d solves he iiil vlue problem y = sec ; y ) = 3 d y) =. 4. Show h y = f) si )) d solves he iiil vlue problem y + y = f), y ) = d y) =. 43. Prove h u f)d) du = 44. Evlue he followig limis b ) lim b fu) u)du.. b) lim d. c) lim rc d. 45. Usig he derivive of hyperbolic fucios d iverse hyperbolic fucios, prove he iegrio formuls give i Tble 7.7 d d Tble 7. i your ebook pges 4 d 45). 4

5 SOLVE THE BELOW EXERCISES FROM YOUR TEXTBOOK. Of course, solve s my eercises s you eed o be sure h you hve lered he coceps d c do compuios wihou error bu I require you o be prepred o solve some of he followig eercises e week: Sec. 4.8 Aiderivives pges 77-8) Eercises:, 4, 7, 8, 4 9, 4, 39 7, 7, 73, 76 9, 98, 4, 3 8,, 5, 7, 8. Sec. 5. Are d Esimig wih Fiie Sums pges 96-98) Eercises:, 4,,. Sec. 5. Sigm Noio d Limis of Fiie Sums pges 34-35) Eercises: 3, 38, 43. Sec. 5.3 The Defiie Iegrl pges 33-37) Eercises: 9, 3, 5, 7, 8, 4 5, 55, 56, 6, 6, 7 86, Sec. 5.4 The Fudmel Theorem of Clculus pges 35-38) Eercises: 7 8, 64, 69 73, Sec. 5.5 Idefiie Iegrls d he Subsiuio Mehod pges ) Eercises: 6, 3 47, 5 66, Sec. 5.5 Subsiuio d Are Bewee Curves pges ) Eercises: 78, 8, 8, 85, 87, 88, 93,, 4, 6, 7, 9 8. Sec. 7. Hyperbolic Fucios pges 45-48) Eercises:, 4, 5, 7,, 3, 5 7, 9, 3 8, 85, 86. We hve lredy see hyperbolic fucios d iverse hyperbolic fucios, heir grphs d derivives. Jus mke review of hem. Sec. 7. The Logrihm Defied s Iegrl pges 49-4) Eercises: 9, 4, 3 7, 33 36, 4, 4, 47, 48, 5, 53 56, 59, 6, 64, 67, 7 Sudy Secio 7. idepedely o ler how oe c defie rigorously he rscedel fucios l d e, d obi ll of heir usul properies. A few weeks ler you hve solved hese eercises, o keep fresh your kowledge i is suggesed o solve he followig eercises from your ebook o pges d 48 43: Quesios o Guide your Review Prcice Eercises Addiiol d Advced Eercises 5