under the curve in the first quadrant.

Transcription

1 NOTES 5: INTEGRALS Nme: Dte: Perod: LESSON 5. AREAS AND DISTANCES Are uder the curve Are uder f( ), ove the -s, o the dom., Prctce Prolems:. f ( ). Fd the re uder the fucto, ove the - s, etwee,.. f ( ) 5. Fd the re uder the curve the frst qudrt.. f ( ) 5. Fd the re uder the fucto, ove the - s, etwee, 7. Heghts of Rectgles (ssume tht the retgles re evely spced) L : heght determed from the left sde of the su-tervl R : heght determed from the rght sde of the su-tervl M : heght determed from the mdpot of the su-tervl AP Clculus AB Chpter 5 Notes Pge

2 Prctce Prolems:. Estmte the re uder the grph of f ( ), from usg three rectgles. Do ths estmto y rght edpots, left edpots d mdpots. Left Edpots Rght edpots Mdpots. Estmte the re uder the grph of f ( ) s from, usg fve rectgles. Do ths estmto y rght edpots, left edpots d mdpots. Left Edpots Rght edpots Mdpots AP Clculus AB Chpter 5 Notes Pge

3 . Estmte the re uder the grph usg the chrt elow. Do ths estmto y rght edpots, left edpots d mdpots. f( ) Left Edpots Rght edpots Mdpots. Fd the dstce trveled the s tervl usg the four methods. Tme (s) Velocty ft / s Left Edpots Mdpots Rght edpots Trpezods Note: The dstce trveled s equl to the re uder the grph of the velocty fucto. AP Clculus AB Chpter 5 Notes Pge

4 LESSON 7.7 TRAPEZOID APPROXIMATION Trpezod Rule (for trpezods wth equl wdths throughout) Are of trpezod: A h( ) f ( ) f ( ) Arecurve f ( ) f ( ) f ( )... f ( ) f ( ) Note: If ot equl wdths, the fd the re of ech trpezod dvdully d sum them up. Prctce Prolems: Appromte the re usg the trpezod rule.. f ( ),,, fd T 5 d T. 7 8 f( ) Note: T N L N R N AP Clculus AB Chpter 5 Notes Pge

5 LESSON 5. THE DEFINITE INTEGRAL The Summto Formuls. c c Emple : 5. ( ) Emple : 6. ( )( ) ( ) Emple : ( ) Emple : 6 Gudeles to fd the re of rego usg Rem Sums Note: REP: A lm f ( ) LEP: A lm f ( ) MP: * lm ( ) A f Defto of Defte Itegrl. Fd the wdth:. Fd the epresso for the edpot: c Note: As, s( ) Are of rego S( ), therefore, we c use ether the LEP or the REP epresso to fd the heghts of the rectgles. c s smpler epresso to use.. Fd the epresso for the heghts: f( c ). Fd the summto of the heghts: f( c ) 5. Multply the Summto of the heghts d the wdth: f ( c ) 6. Tke the lmt s : lm f ( c ) * lm f ( ) f ( ) AP Clculus AB Chpter 5 Notes Pge 5

6 Prctce Prolems: Use the Rem Sum to ppromte the re of fucto.. from, f ( ). 5 AP Clculus AB Chpter 5 Notes Pge 6

7 Deftos of Two Specl Defte Itegrls Addtve Itervl Property Propertes of Defte Itegrls Preservto of Iequlty. If f s defed t, Emples: the f ( ). If f s tegrle o,, the f ( ) f ( ) If f s tegrle o the three closed tervls determed y,, d c, the c f ( ) f ( ) f ( ) c.. Emple: If f d g re tegrle o, d k s costt, the the fuctos of kf d f g,, d re tegrle o kf k f. k k ( ). ( ) ( ) f ( ) g( ) f ( ) g( ).. If f s tegrle d oegtve o the closed tervl,, the ( ). f. If f d g re tegrle o the closed tervl, d f ( ) g( ) the f ( ) g( ) for every,, Grph: Prctce Prolems: Evlute y terpretg terms of res.. y. 6 AP Clculus AB Chpter 5 Notes Pge 7

8 LESSON 5. THE FUNDAMENTAL THEOREM OF CALCULUS Prt : The Fudmetl Theorem of Clculus (Are Fucto) Prt : The Fudmetl Theorem of Clculus The Secod Fudmetl Theorem of Clculus If f s cotuous o,, the the fucto g defed y g( ) f ( t) dt s cotuous o, d dfferetle o,, d g '( ) f ( ) If fucto f s cotuous o the closed tervl, d F s tdervtve of f o the tervl,, the Are f ( ) F( ) F( ) If f s cotuous o ope tervl I cotg, the, for every the d tervl, f ( t) dt f ( ) Note: Ths theorem s oly e used whe sked to fd F'( ) d the lower oud s costt. Emple: Emple: Evlutg Defte Itegrl Proof: u du u Prctce Prolems: Evlute.. d d t t 7 e dt. s t 5 7 t dt Proof: AP Clculus AB Chpter 5 Notes Pge 8

9 d t dt cos. ( ) 5 e. sec f t dt, fd f '( ) t t dt d 8. Gve g( ) t dt. Fd g (). 7. csc cot 6 9. Gve f '( ) 6 6. If f 5, the f?. A prtcle moves log the -s so tht t y tme t, ts ccelerto s gve y t ( ) l t. If the velocty of the prtcle s t t, the the velocty of the prtcle t tme t s? AP Clculus AB Chpter 5 Notes Pge 9

10 WARM-UP: IMPORTANT ON AP EXAM. e. 5 d. 5 t 7t dt. t 7 d t dt 5. d cos t d te dt 6. 7 l t t dt 7. d t st 5 57 dt 8. sect 6 l d dt AP Clculus AB Chpter 5 Notes Pge

11 LESSON 5. INDEFINITE INTEGRALS AND THE NET CHANGE THEOREM Itegrl Ides. If f s defed t, Itegrto of Eve d Odd Fuctos Let f e tegrle o the closed tervl,. the f ( ). If f s tegrle o,, the f ( ) f ( ) If f s eve fucto, the f ( ) f ( ) If f s odd fucto, the f ( ) Idefte Itegrl F ( ) s the tdervtve of f( ) f ( ) F '( ) F( ) F '( ) f ( ) d F ( ) f ( ) F '( ) Importt Note: A defte tegrl s umer (re uder the curve). A defte tegrl s fmly of fuctos. Tle of Idefte Itegrl cf ( ) c f ( ) k k C f ( ) g( ) f ( ) g( ) C, l C e e C cos s C s cos C sec csc t C cot C sec t sec C csc cot csc C s C t C sh cosh C cosh sh C AP Clculus AB Chpter 5 Notes Pge

12 The Net Chge Theorem F '( ) F ( ) F ( ) Qutty, mout ccummulted the tervl of dom rte Prctce Prolems: Evlute.. The tegrl c e rewrtte s: The verte s:,,, Gve: vt () Dsplcemet = et chge = v () t dt Dstce trveled: speed dt () v t dt. v( t) t t. Fd:. The dsplcemet o,. The dstce trveled o,. f( ) represets the rte whch freshme were leded Mr. Shy s morg smoothes freshme/secod. Wht does f ( ) represet? AP Clculus AB Chpter 5 Notes Pge

13 LESSON 5.5 THE SUBSTITUTION RULE The Susttuto Rule Let g e fucto whose rge s tervl I, d let f e fucto tht s cotuous o I. If g s dfferetle o ts dom d F s tdervtve of f o I, the f ( g( )) g '( ) F( g( )) C If u g( ), the du g '( ) f ( u) du F( u) C d If g s dfferetle fucto of, the g ( ) g( ) g '( ) C, Equvletly, f u g( ), the u u du C, Prctce Prolems: Evlute the followg prolems cos e 6. f ( t) t 7, F() 5 where F '( t) f ( t). Fd F (). AP Clculus AB Chpter 5 Notes Pge

14 7. t Other Itegrl Formuls cot l s C sec l sec t C csc l csc cot C t l sec C l cos C Prctce Prolems: Evlute the followg prolems. 9. sec. 9 Emples:.. 5. t5. cot 6 AP Clculus AB Chpter 5 Notes Pge