0 dx C. k dx kx C. e dx e C. u C. sec u. sec. u u 1. Ch 05 Summary Sheet Basic Integration Rules = Antiderivatives Pattern Recognition Related Rules:

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1 Ch 05 Smmry Sheet Bic Itegrtio Rle = Atierivtive Ptter Recogitio Relte Rle: ( ) kf ( ) kf ( ) k f ( ) Cott oly! ( ) g ( ) f ( ) g ( ) f ( ) g( ) f ( ) g( ) Bic Atierivtive: C 0 0 C k k k k C Mlt. S. 1 C, 1 A 1 Divie 1 Atierivtive Trig. Fctio (memory pt hit) Wtch the ig! i co co i C co i i co C ee the egtive! t ec ec t C ec ec t ec t ec C cot cc cc cot C cc cc cot cc cot cc C Atierivtive Epoetil & Logrithmic Fctio e e e e C l 1 C l 1 1 l, 0 l C olte vle mke > 0 Atierivtive Ivere Trig. Fctio > 0, i ifferetile fctio of. 1 1 Sice the erivtive of co i jt the oppoite of i, etc yo ee oly to e oe memer from ech pir. 1 1 i i C t 1 t C 1 1 ec ec 1 C 1 Pge 1 of 6 S. Stirlig

2 Ch 05 Smmry Sheet Itegrtio Techiqe U-tittio AKA Atiifferetitio of Compoite Let g e fctio whoe rge i itervl I, let f e fctio tht i cotio o I. If g i ifferetile o it omi F i tierivtive of f o I, the f ( g ( )) g ( ) F ( g ( )) C OR If g( ), the g( ) f ( ) F( ) C Write itegr i term of Thm 5.14 Chge of Vrile for Defiite Itegrl If the fctio g( ) h cotio erivtive o the cloe itervl [, ] f i cotio o the rge of g, the Itegrtio Propertie Propertie for Defiite Itegrl f g re itegrle fctio o [, ]. Zero: f ( ) 0 Defiitio (o re) Thik re! Orer of Itegrtio: f ( ) f ( ) Symmetric limit: If f i o, f ( ) 0 Defiitio (ice egtive, revere to ) If f i eve, Zero: re ove = re elow. f ( ) f ( ) y-i ymmetry: ole the re from 0. Cott Mltiple: kf ( ) k f ( ) Sm & Differece: 0 Ay mer k, eve k = 1. f ( ) g( ) f ( ) g( ) (Like erivtive.) c c Aitivity: f ( ) f ( ) f ( ) Domitio: f ( ) g( ) o [, ] f ( ) g( ) or Preervtio of Ieqlity g( ) f ( g( )) g( ) f ( ) g ( ) If -tittio i e, chge the limit of itegrtio to lo. f( ) 0 o [, ] f ( ) 0 Some Gielie 1. Ptter recogitio. If etr tff i the itegr. Try -tittio. (Let = iermot fctio, rewrite i term of. Itegrte. 3. Try lterig the itegr: Ue trig. ietity Mlt. & iv. y the me # Seprte ito itegrl Ue property of itegrl Simplify the itegr Log iviio 4. Give p wit for Clc BC! Are ove -i po. Are elow -i eg. If weepig ot re left to right, poitive If weepig ot re right to left, egtive Pge of 6 S. Stirlig

3 Ch 05 Smmry Sheet Slope Fiel Give geerl hpe of geerl oltio. C e whe tierivtive c t e fo. Solve ifferetil eqtio y f( ) y f ( ) y ( ) f Geerl oltio F( ) Applictio: (ee p5.) Ie. Give ccelertio fi poitio. Nee iitil coitio. Uig itegrtio to work ckwr. C fmily of fctio Prticlr oltio with iitil coitio F( ) # pecific fctio Atierivtive oig ifferetitio Wht wol yo erive to get? F tierivtive of f if F( ) f ( ). Alo (i other ottio), f ( ) f ( ) C or vi ver f ( ) f ( ) Fmetl Theorem of Clcl: Atierivtive Prt (ometime clle the FTC) Give re fctio A f tht weep ot re er f(t), A ( ) f ( t) t. f The rte t which the re i eig wept ot i eql to the height of the origil fctio. Bece the rte i the erivtive, the erivtive of the re fctio eql the origil fctio: Af ( ) f ( t) t f ( ) Upper limit fctio? FTC Atierivtive prt with chi rle: g( ) f () t t = f ( g( )) g( ) c Fmetl Theorem of Clcl: Evltio Prt (ometime clle the 1 t FTC) Let F e y tierivtive of the cotio fctio f; the f ( ) F( ) F( ). OR Upper lower limit fctio? Rewrite y itivity & FTC Atierivtive prt with chi rle: g( ) f () t t h( ) h( ) g ( ) = f ( t) t f ( t) t # # = f ( h( )) h( ) + f ( g( )) g( ) Upper lower limit fctio? Ue FTC Evltio Prt F() i tierivtive of f. g( ) f () t t = F ( g ( )) F ( h ( )) h( ) Pge 3 of 6 S. Stirlig

4 Ch 05 Smmry Sheet Are 1) Ue Geometric Forml (Propertie) if poile Rectgle, trigle, trpezoi, emicircle, ) Approimtio Metho Mi cocept: Fill i the figre with rectgle. Icree ccrcy: LRAM v RRAM (epe o fctio icreig or ecreig) MRAM Trpezoi Icree ccrcy: icree, 0 ll metho ecome the efiite itegrl. Rectgle: f() height, with Prtitio the itervl [, ] ito itervl: Are lim f(epoit) i 1 ( ) ( 1) LRAM S f i i1 RRAM S( ) f i i1 eiet to e! Smmtio: k 1 1 i1 MRAM e mipoit of the egmet for the height: S( ) f i1 Trpezoil Rle f ( 0) f ( 1) f ( )... f ( 1) f ( ) ote: mt e the me. k 1... ki k i ice k i cott i1 i1 i every term. i i i i i1 i1 i1 Sm of eqece [STAT] MATH 5:m( [STAT] OPS 5:eq( the forml, vrile, trt, e. 3) Itegrtio Pge 4 of 6 Defiitio of the Are of Regio i the Ple Let f e cotio oegtive o the itervl [, ]. The re of the regio oe y the grph of f, the -i, the verticl lie = = i Are lim f ( ci ), i 1 c i i where. i 1 Defiitio of Defiite Itegrl y if mke ifiite mer of prtitio the prtitio 0, the the pproimtio m = the efiite itegrl. Thm 5.5 The Defiite Itegrl the Are of Regio If f i cotio oegtive o the cloe itervl [, ], the the re of the regio oe y the grph of f, the -i, the verticl lie = = i give y. Are = f ( ) Riem Sm I geerl, the prtitio, o ot ee to e the me. It i eier to clclte whe they re the me. Defiite Itegrl: Are ove the -i poitive itegrl Are elow the -i egtive itegrl Ue FTC Evltio Prt f ( ) = F( ) F( ) OR y of the previo re fiig metho, geometry. S. Stirlig

5 Ch 05 Smmry Sheet Are Accmltio Fctio Are Net Chge The emple e velocity poitio; however, the followig wol hol for ANY give rte of chge. The re ccmltio fctio: A( ) f ( t) t where i the trtig -vle move log the -i to ccmlte re er f(t). Differetite ( ) ( ) v( ) ( ) v( ) ( ) Give the et chge i the tierivtive. (Ie. The fctio give the et chge i poitio whe the itegr i velocity.) y Itegrte By FTC Evltio: How fr hve yo trvele o itervl? f ( t) t F( ) F( ) et chge i poitio Wht i yor crret poitio? F( ) F( ) f ( t) t mile per hor Are = mile hor mile hor Net chge i mile. hor Totl mile trvele = crret poitio Strtig poitio (mile) Net chge i mile = itce trvele o itervl (, ). By FTC Atierivtive: How ft re yo trvelig? Tke the erivtive. A( ) f ( t) t f ( ) The rte t which yo ccmlte the re (the et chge i poitio) epe o the height of the velocity fctio. So F( ) f ( ), where f() i velocity fctio F() i poitio. Comprig Procee How re the iefiite itegrl the efiite itegrl relte? A re ccmltio fctio, y trtig t time of 3: 3 3 (3) 6 y FTC Evltio A iefiite itegrl, with iitil coitio of 0 (ice o re h ee ccmlte t the trtig poit) whe = 3. C the 0 (3) C the tierivtive i 6., 6 C Pge 5 of 6 S. Stirlig

6 Ch 05 Smmry Sheet Mic. Thm Differetiility implie Cotiity which implie Itegrility. (Yo c itegrte ythig tht yo c fi the re er the grph.) Thm 5.10 Me Vle Theorem for Itegrl If f i cotio o the cloe itervl [, ], the there eit mer c i the cloe itervl [, ] ch tht f ( ) f ( c)( ) MVT for Itegrl Fi rectgle tht h the me re the efiite itegrl. Solve for f(c) Averge f() vle Defiitio of the Averge Vle of Fctio o Itervl If f i itegrle o the cloe itervl [, ], the the verge vle of f o the itervl i 1 f ( ) f ( c) Averge Vle Fi the et poile pproimtio for the verge of the cotio fctio vle.. Thi i ot meric verge! Clcltor: From the home cree: [MATH] 9:fIt(fctio, vrile, lower, pper, tolerce) evlte the efiite itegrl tolerce i optiol t c pee p the proce. To get grph of itegrl efie fctio From grph of fctio: [CALC] 7: f ( ) the type i the lower limit? the pper limit? the clcltor retr the efiite itegrl, fill i the re. Y1 f ( t) t Y1 = [MATH] 9:fIt(fctio i term of T, T, trt, X) Pge 6 of 6 S. Stirlig