( ) Ú Ú. Ú Ú, c is a constant Ú Ú Ú Ú Ú. a Ú for any value of c. Ú Ú Ú Ú Ú Ú Ú ( ) =- ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

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1 Definie Inegrl: Suppose f on [, ]. Divie [, ] wih D n hoose Clulus Che Shee Inegrls Definiions is oninuous ino n suinervls of from eh inervl. * i * Then limâ ( i ) f = f D. næ i = AniDerivive : An nierivive of f ( ) is funion, F( ), suh h F ( ) = f ( ). Inefinie Inegrl : f ( ) = F( ) where F( ) is n nierivive of f ( ). Funmenl Theorem of Clulus f is oninuous on [, ] hen Vrins of Pr I : u g( ) = f () is lso oninuous on [, ] f () u ( ) f u( ) = È Î g = f = f. f () v ( ) f v( ) = È v Î f is oninuous on[, ], F( ) is u f () u f u v f = v F = f ) Pr I : If n () Pr II : n nierivive of f ( ) (i.e. f F F. hen = ± = ± ± = ± f g f g f g f g 0 f = = f f = f f f for ny vlue of. If f ( ) g( ) on hen If f ( ) 0 on hen 0 Properies f = f [ ] [ v ], is onsn f ( ) = f ( ), is onsn f g f = f f If m f ( ) M on hen m ( ) f ( ) M ( ) n n = n = n, = = ln = ln lnuu = uln ( u) u u u e u e = Common Inegrls osuu = sin u Visi hp://uoril.mh.lmr.eu for omplee se of Clulus noes. sinuu = osu se uu = n u seunuu = seu suouu = su s uu = o u nuu = ln seu seuu = ln seu n u u u = u n u u = sin u 00 Pul Dwins

Funmenl Theorem of Clulus f is oninuous on [, ] hen Vrins of Pr I : u g( ) = f () is lso oninuous on [, ] f () u ( ) f u( ) = È Î g = f = f.

2 Clulus Che Shee Snr Inegrion Tehniques Noe h mny shools ll u he Susiuion Rule en o e ugh in Clulus II lss. u Susiuion : The susiuion u = g( ) will onver u = g. For inefinie inegrls rop he limis of inegrion. E. os u = fi u = fi = u :: 8 = fi u = = = fi u = = Inegrion y Prs : uv= uv vu n g f ( g ) g ( ) = f ( u) u using g = 8 = sin( u) = ( sin( 8) sin() ) 8 inegrl n ompue u y iffereniing u n ompue v using v E. u = v= e fi u = v=e e = e e =e e os os u u uv = uv vu. Choose u n v from E. ln = v. u = ln v= fi u = v= ( ) ln= ln = ln = ln ln Prous n (some) Quoiens of Trig Funions n m n m For sin os we hve he following : For n se we hve he following :. n o. Srip sine ou n onver res o osines using sin = os, hen use he susiuion u = os.. m o. Srip osine ou n onver res o sines using os = sin, hen use he susiuion u = sin.. n n m oh o. Use eiher. or. 4. n n m oh even. Use oule ngle n/or hlf ngle formuls o reue he inegrl ino form h n e inegre.. n o. Srip ngen n sen ou n onver he res o sens using n = se, hen use he susiuion u = se.. m even. Srip sens ou n onver res o ngens using se = n, hen use he susiuion u = n.. n o n m even. Use eiher. or. 4. n even n m o. Eh inegrl will e el wih ifferenly. sin sin os os os sin = os Trig Formuls : =, =, E. n se = 4 ( se se ) nse 4 ( u ) uu ( u se ) 4 n se n se n se = = = = se se 7 7 E. sin os 4 sin sin sin (sin ) sin = = os os os (os ) sin = = os (u) u u4 = u = u u ( os ) u u = se ln os os Visi hp://uoril.mh.lmr.eu for omplee se of Clulus noes. 00 Pul Dwins

os u = fi u = fi = u :: 8 = fi u = = = fi u = = Inegrion y Prs : uv= uv vu n g f ( g ) g ( ) = f ( u) u using g = 8 = sin( u) = ( sin( 8) sin() ) 8 inegrl n ompue u y iffereniing u n ompue v using v

3 Clulus Che Shee Trig Susiuions : If he inegrl onins he following roo use he given susiuion n formul o onver ino n inegrl involving rig funions. fi = sinq os q = sin q fi = seq n q = se q fi = nq se q = n q 6 E. 49 sin = q fi = osq q 4 4sin 4os 4 9 = q = q = os q Rell =. Beuse we hve n inefinie inegrl we ll ssume posiive n rop solue vlue rs. If we h efinie inegrl we nee o ompue q s n remove solue vlue rs se on h n, Ï if 0 = Ì Ó if < 0 In his se we hve 4 9 = osq. Û ı 6 = 4 sin q q sin q 9 ( os ) ( os ) q q q s o = q = q Use Righ Tringle Trig o go o s. From susiuion we hve sinq = so, From his we see h o 49 q =. So, = Pril Frions : If inegring P where he egree of Q P is smller hn he egree of Q( ). For enominor s ompleely s possile n fin he pril frion eomposiion of he rionl epression. Inegre he pril frion eomposiion (P.F.D.). For eh for in he enominor we ge erm(s) in he eomposiion oring o he following le. For in Q( ) Term in P.F.D For in Q( ) A A B ( ) ( ) Term in P.F.D A A A A B A B 7 ( )( 4) E = ( )( 4) 4 = ( ) = 4ln ln 4 8n Here is pril frion form n reomine. A ( A B C 4) ( B C)( ) = = Se numerors equl n olle lie erms. 7 = A B C B 4A C Se oeffiiens equl o ge sysem n solve o ge onsns. A B= 7 C B= 4A C = 0 A= 4 B= C = 6 An lerne meho h someimes wors o fin onsns. Sr wih seing numerors equl in 7 = A 4 B C. Chose nie vlues of n plug in. previous emple : For emple if = we ge 0 = A whih gives A = 4. This won lwys wor esily. Visi hp://uoril.mh.lmr.eu for omplee se of Clulus noes. 00 Pul Dwins

If we h efinie inegrl we nee o ompue q s n remove solue vlue rs se on h n, Ï if 0 = Ì Ó if < 0 In his se we hve 4 9 = osq.

4 f Clulus Che Shee Appliions of Inegrls Ne Are : ( ) represens he ne re eween f n he is wih re ove is posiive n re elow is negive. Are Beween Curves : The generl formuls for he wo min ses for eh re, Èupper funion È Îlower funion Î & = fi = Îrigh funion Îlef funion y = f fi A= È È f y A If he urves inerse hen he re of eh porion mus e foun iniviully. Here re some sehes of ouple possile siuions n formuls for ouple of possile ses. A= f ( y) g( y) = A= f g A f g g f Volumes of Revoluion : The wo min formuls re V = A( ) n V = A y. Here is some generl informion ou eh meho of ompuing n some emples. Rings Cyliners A = p ( ( ouer rius) ( inner rius) ) A = p ( rius) ( wih / heigh) imis: /y of righ/o ring o /y of lef/op ring imis : /y of inner yl. o /y of ouer yl. f, f y, f y, f, Horz. Ais use g( ), A( ) n. Ver. Ais use g( y ), A( y ) n. Horz. Ais use g( y ), A( y ) n. Ver. Ais use g( ), A( ) n. E. Ais : y = > 0 E. Ais : y = 0 E. Ais : y = > 0 E. Ais : y = 0 ouer rius : f ( ) inner rius : g( ) ouer rius: g( ) inner rius: f ( ) rius : y wih : f ( y) g( y) rius : y wih : f ( y) g( y) These re only few ses for horizonl is of roion. If is of roion is he is use he y = 0 se wih = 0. For veril is of roion ( = > 0 n = 0 ) inerhnge n y o ge pproprie formuls. Visi hp://uoril.mh.lmr.eu for omplee se of Clulus noes. 00 Pul Dwins

mus e foun iniviully. Here re some sehes of ouple possile siuions n formuls for ouple of possile ses.

5 Wor : If fore of F( ) moves n oje in, he wor one is W = Clulus Che Shee F Averge Funion Vlue : The verge vlue of f ( ) on is fvg = f ( ) Ar engh Surfe Are : Noe h his is ofen Cl II opi. The hree si formuls re, = s SA= p ys (roe ou is) SA= p s (roe ou yis) where s is epenen upon he form of he funion eing wore wih s follows. ( ) s = if y = f, s = if = f y, y () () s = if = f, y = g, r s = r q if r = f q, q Wih surfe re you my hve o susiue in for he or y epening on your hoie of s o mh he ifferenil in he s. Wih prmeri n polr you will lwys nee o susiue. Improper Inegrl An improper inegrl is n inegrl wih one or more infinie limis n/or isoninuous inegrns. Inegrl is lle onvergen if he limi eiss n hs finie vlue n ivergen if he limi oesn eis or hs infinie vlue. This is ypilly Cl II opi. Infinie imi. lim f = f ( ). f ( ) = lim f ( ) Æ. = Æ q Æ f f f provie BOTH inegrls re onvergen. Disoninuous Inegrn. Dison. : f ( ) = lim f ( ). Dison. : f ( ) = lim f ( ). Disoninuiy < < : = Æ f f f provie oh re onvergen. Comprison Tes for Improper Inegrls : If f ( ) g( ) 0 on [, ) f hen,. If onv. hen onv.. If ivg. hen Useful f : If > 0 hen For given inegrl f ( ) ivie [, ] g p g onverges if p > n iverges for p. f ivg. Approiming Definie Inegrls n n (mus e even for Simpson s Rule) efine D = n n ino n suinervls [, ], [, ],, [ ] 0 wih 0 n, n * * * Mipoin Rule : ªD È ( ) ( n) = n n = hen, f f f f * Î, i i, i D f ª Èf 0 f f f n f n Î D f ª Èf 0 4f f f 4f f Î is mipoin [ ] Trpezoi Rule : Simpson s Rule : n n n Visi hp://uoril.mh.lmr.eu for omplee se of Clulus noes. 00 Pul Dwins

( ) s = if y = f, s = if = f y, y () () s = if = f, y = g, r s = r q if r = f q, q Wih surfe re you my hve o susiue in for he or y epening on your hoie of s o mh he ifferenil in he s.

Improper Integrals. Dr. Philippe B. laval Kennesaw State University. September 19, 2005. f (x) dx over a finite interval [a, b].

Improper Integrals. Dr. Philippe B. laval Kennesaw State University. September 19, 2005. f (x) dx over a finite interval [a, b]. Improper Inegrls Dr. Philippe B. lvl Kennesw Se Universiy Sepember 9, 25 Absrc Noes on improper inegrls. Improper Inegrls. Inroducion In Clculus II, sudens defined he inegrl f (x) over finie inervl [,