AP Calculus Notes: Unit 6 Definite Integrals. Syllabus Objective: 3.4 The student will approximate a definite integral using rectangles.

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1 AP Clculus Notes: Uit 6 Defiite Itegrls Sllus Ojective:.4 The studet will pproimte defiite itegrl usig rectgles. Recll: If cr is trvelig t costt rte (cruise cotrol), the its distce trveled is equl to rte time. O grph, this would look like the oe show, for cr trvelig t 6 mph for hours. v Rte The re of the shded regio = 6 = miles; the totl distce trveled i hours. Fidig Distce Trveled Whe Velocit Vries To estimte the re of irregulr regio, we will estimte usig rectgles formed suitervls. d moves log the -is with velocit E: A prticle strts t v t t for time t. Where is the prticle t t? Use 8 suitervls. (We will use midpoit rectgles.) v Time t Sueritervls: i Midpoits m : Heights = m i :, , 4 8, Widths = 4 Pge of Perso Pretice Hll 7 Clculus: Grphicl, Numericl, Algeric These otes re liged to the tetook refereced ove d to the College Bord Clculus AB curriculum Ares = m i : Are So the prticle is close to 4 whe t. Rectgulr Approimtio Methods (RAM): Used to pproimte the re uder curve t

2 AP Clculus Notes: Uit 6 Defiite Itegrls LRAM: Left-Hd Rectgles Use the left edpoits of ech itervl to drw the heights of ech rectgle. MRAM: Midpoit Rectgles Use the midpoits of ech itervl to drw the heights of ech rectgle. RRAM: Right-Hd Rectgles Use the right edpoits of ech itervl to drw the heights of ech rectgle. E: Approimte the re uder the curve pproimtio methods listed ove. from to usig 6 suitervls. Use ll of the LRAM Are.75 LRAM RRAM Are.875 RRAM Pge of Perso Pretice Hll 7 Clculus: Grphicl, Numericl, Algeric These otes re liged to the tetook refereced ove d to the College Bord Clculus AB curriculum.

3 AP Clculus Notes: Uit 6 Defiite Itegrls MRAM 5 7 Are RRAM Note: To get etter pproimtio of the re, more rectgles (smller itervls) c e used. Techer Note: There re progrms for the TI-84 d TI-89 for pproimtig re usig RAM. These progrms will llow ou to estimte the re usig hudreds of rectgles! Check the techolog resource guide give to ou with our tetook to fid progrms. Eplortio: Which RAM is the iggest? Estimte the re uder the grph of e i the itervl,. Usig ll three RAM. List them from smllest to gretest. Estimte the re uder the grph of i the itervl,. Usig ll three RAM. List them from smllest to gretest. Drw coclusio usig properties of the grphs ove s to which RAM is the iggest. Test our coclusio with few grphs tht ou choose o our ow sed upo the properties i our coclusio. Solutio: If curve is icresig, the RRAM is the iggest ecuse the rectgles go ove the curve, d LRAM is the smllest ecuse the rectgles lie elow the curve. If curve is decresig, the LRAM is the iggest ecuse the rectgles go ove the curve, d the RRAM is the smllest ecuse the rectgles lie elow the curve. I oth cses, the MRAM is somewhere i etwee the RRAM d LRAM. I curves tht re ot strictl mootoic, such s si,, it cot e determied. o the itervl Pge of Perso Pretice Hll 7 Clculus: Grphicl, Numericl, Algeric These otes re liged to the tetook refereced ove d to the College Bord Clculus AB curriculum.

4 AP Clculus Notes: Uit 6 Defiite Itegrls You Tr: The tle elow shows the velocit of cr t secod time itervls. Estimte how fr the cr trveled usig LRAM d RRAM sums. Time (sec) Velocit (ft/sec) QOD: Give emple of fuctio (d itervl) for which the RRAM d LRAM re equl o tht itervl. Pge 4 of Perso Pretice Hll 7 Clculus: Grphicl, Numericl, Algeric These otes re liged to the tetook refereced ove d to the College Bord Clculus AB curriculum.

5 AP Clculus Notes: Uit 6 Defiite Itegrls Sllus Ojective:.5 The studet will evlute defiite itegrl s the limit of Reim sum.. The studet will fid the re etwee two or more curves (etwee curve d -is). Recll: Sigm Nottio = sum k... k Voculr d Nottio: Below is the grph of fuctio f. The itervl, prtitios: P,,,...,, hs ee prtitioed ito suitervls, clled (Note Ech suitervl is ot ecessril the sme width.) The width of the k th rectgle is equl to k. Ech rectgle is creted choosig vlue, c, withi ech suitervl d drwig rectgle with height of f c. = c - c c = Nottio: P = legth of suitervl (clled the orm) Riem Sum (sum of the re of rectgles): S f c k k k Note: Smller prtitios crete more rectgles. Defiitio: Let f e fuctio defied o,. For prtitio P of,, let the umers c k e chose ritrril i the suitervls k, k. If there eists umer I such tht lim f ck k I, the f is itegrle o, defiite itegrl of f over,. P k d I is the Pge 5 of Perso Pretice Hll 7 Clculus: Grphicl, Numericl, Algeric These otes re liged to the tetook refereced ove d to the College Bord Clculus AB curriculum.

6 AP Clculus Notes: Uit 6 Defiite Itegrls Theorem: All cotiuous fuctios re itegrle. Defiite Itegrl: Let f e cotiuous o,, d let, e prtitioed ito suitervls of equl legth. The the defiite itegrl of f over, is lim f c. k k k k P k Note: The lim f c mes tht the legth of the prtitios re pprochig (gettig smller). This is the sme ide s lim f c, which mes tht the umer of rectgles re pprochig. k k k k Nottio for Defiite Itegrls: lim Note tht the itegrl smol resemles S, ecuse itegrl is sum. f c f d, red the itegrl from to of f of d. Voculr of Defiite Itegrls: Itegrl Sig Upper Limit of Itegrtio Lower Limit of Itegrtio f Itegrd d Vrile of Itegrtio (dumm vrile) The itervl k 6 E: The itervl 4, is prtitioed ito suitervls of equl legth. Let m k deote the midpoit of the k th suitervl. Epress the limit 4, represets,. (This is wh m represets vlue of i ech suitervl. So Itegrl: d Pge 6 of Perso Pretice Hll 7 Clculus: Grphicl, Numericl, Algeric These otes re liged to the tetook refereced ove d to the College Bord Clculus AB curriculum. mk mk s itegrl. k lim ) f 4 7.

7 AP Clculus Notes: Uit 6 Defiite Itegrls Are Uder Curve (ove the -is): If uder the curve f is oegtive d itegrle over, f from to is the itegrl of f from to : f d Note: We c use itegrls to clculte res, d use res to clculte itegrls!, the the re E: Evlute the itegrl Grph the fuctio 9 d. 9. Shde the regio i the itervl,. Fid the re of the shded regio: The figure shded is ¼ of circle with rdius of. r 9 Are =. So, d 4 Questio: Wht if fuctio is opositive? The vlue of the fuctio represetig the heights of the rectgles will ll e egtive. So the heights of the rectgles would e the opposite of the fuctio vlue. Are of Regio Betwee Curve d the -Ais (uder the -is): If over,, the the re etwee the curve from to : Are = f d f is opositive d itegrle f d the -is from to is the opposite of the itegrl of f f with oth positive d Are of Regio Betwee Curve d the -Ais: For itegrle fuctio egtive vlues o itervl,, the totl re etwee the curve d the -is is equl to the Riem sum of f over the itervl the fuctio is ove the -is mius the Riem sum of f itervl the fuctio is elow the -is. over the Defiite Itegrl: For itegrle fuctio, f d = (re ove the -is) (re elow the -is). Note: A itegrl c e egtive, re cot! Pge 7 of Perso Pretice Hll 7 Clculus: Grphicl, Numericl, Algeric These otes re liged to the tetook refereced ove d to the College Bord Clculus AB curriculum.

8 AP Clculus Notes: Uit 6 Defiite Itegrls E: Use res to fid the vlue of Sketch the grph d shde the regio. 5 d. The re of the shded regio is equl to A lw4. So the vlue of 5 d. (The swer is egtive ecuse the fuctio f lies elow the -is. Use this emple to epli the followig formul. The Itegrl of Costt Fuctio: If f d cd c. f c, where c is costt, o the itervl,, the Clcultig Defiite Itegrls: O the TI-84, defiite itegrl is pproimted usig umericl itegrtio, NINT. The kestrokes re fit(fuctio, vrile of itegrtio, lower limit, upper limit). fit c e foud i the Mth meu. E4: Evlute 7 5 d. 7 5 d 6.64 Now check our swers to the previous emples usig the clcultor. Pge 8 of Perso Pretice Hll 7 Clculus: Grphicl, Numericl, Algeric These otes re liged to the tetook refereced ove d to the College Bord Clculus AB curriculum.

9 AP Clculus Notes: Uit 6 Defiite Itegrls Note: All cotiuous fuctios re itegrle. A discotiuous fuctio MAY e itegrle. E5: Evlute the itegrl: Sketch the grph d shde the regio. 4 d Are ove the -is = () = Are elow the -is = 4() = 4 d = (re ove the -is) (re elow the -is) = 4 4 You Tr: Evlute the itegrl 8 5 d usig res. Check our swer o the clcultor. 5 QOD: Give emple of fuctio d itervl o which the fuctio is NOT itegrle. Pge 9 of Perso Pretice Hll 7 Clculus: Grphicl, Numericl, Algeric These otes re liged to the tetook refereced ove d to the College Bord Clculus AB curriculum.

10 AP Clculus Notes: Uit 6 Defiite Itegrls Sllus Ojective:.7 The studet will solve prolems usig the properties of defiite itegrls.. The studet will fid the verge vlue of fuctio o itervl. RULES FOR DEFINITE INTEGRALS f d f d f Epltio: The width of the suitervls will e, which is the opposite of. d Epltio: Itegrtig from to will crete o regio uder the curve, which will hve re of. k f d k f d, for costt k Epltio: The re will e multiplied the costt. f g d f d g d Epltio: This follows from the rules of Riem Sums. c c f d f d f d Epltio: Accumultio of re over two itervls. m mi f f d f Epltio: The vlue of the et re will e etwee the siged re of the smllest rectgle d the siged re of the lrgest rectgle. If f g for, the f d g d. Pge of Perso Pretice Hll 7 Clculus: Grphicl, Numericl, Algeric These otes re liged to the tetook refereced ove d to the College Bord Clculus AB curriculum.

11 AP Clculus Notes: Uit 6 Defiite Itegrls E: Use the properties of itegrls to evlute the followig, give 5 f d 6, g d, h d, f d 8. g hd g h d g d h d 5. f d 5. g 5 f d f d 6 5 h d Not possile with the iformtio give; o product rule for itegrtio 5 4. f d 5. 4 f g d 5 5 f d f d f d 86 4f g d 4 f d g d si d. E: Fid the upper d lower ouds of the itegrl Upper oud: m Lower oud: f f mi si d. So, Checkig o the clcultor, we hve si d, which is withi our ouds. Averge Vlue of Fuctio: If f is itegrle o,, its verge (me) vlue o, is v f f d Thik out it: To fid the verge (me) of set of umers, we dd the umers d divide how m vlues there re. The itegrl is ifiite sum. So we re ddig ll of the fuctio vlues d dividig the legth of the itervl. Pge of Perso Pretice Hll 7 Clculus: Grphicl, Numericl, Algeric These otes re liged to the tetook refereced ove d to the College Bord Clculus AB curriculum.

12 AP Clculus Notes: Uit 6 Defiite Itegrls E: Fid the verge vlue of f o the itervl t some poit i the give itervl?, 4. Does the fuctio tke o this vlue 4 v f d Does the fuctio tke o this vlue? ; Yes, 7 is i the itervl 8 7, 4. Me Vlue Theorem for Defiite Itegrls: If f is cotiuous o,, the t some poit c i,, f c f d You Tr: Let 6 f d 9 d f d 5. Wht is the vlue of f d? 6 QOD: Descrie how fidig the verge vlue of fuctio reltes to fidig the verge vlue of group of umers. Smple AP Clculus AB Em Questio(s): B = f () C A. The regios A, B, d C i the figure ove re ouded the grph of the fuctio f d the -is. If the re of ech regio is, wht is the vlue of (A) (B) (C) 4 (D) 7 (E) Pge of Perso Pretice Hll 7 Clculus: Grphicl, Numericl, Algeric These otes re liged to the tetook refereced ove d to the College Bord Clculus AB curriculum. f d?

13 AP Clculus Notes: Uit 6 Defiite Itegrls t t. The velocit, i ft/sec, of prticle movig log the -is is give the fuctio vt e te. Wht is the verge velocit of the prticle from time t to time t? (A).86 ft/sec (B) ft/sec (C).89 ft/sec (D) 4.67 ft/sec (E) 79.4 ft/sec. O the closed itervl,4, which of the followig could e the grph of fuctio f with the propert 4 tht f t dt 4? (A) (B) (C) (D) (E) Pge of Perso Pretice Hll 7 Clculus: Grphicl, Numericl, Algeric These otes re liged to the tetook refereced ove d to the College Bord Clculus AB curriculum.

14 AP Clculus Notes: Uit 6 Defiite Itegrls Sllus Ojectives:.8 The studet will evlute defiite itegrls usig the Fudmetl Theorem of Clculus..9 The studet will iterpret geometricll the Fudmetl Theorem of Clculus.. The studet will determie the tiderivtive of elemetr fuctio. The Fudmetl Theorem of Clculus (Prt ): If f is cotiuous o hs derivtive t ever poit i df d,, d f t dt f Note: The lower limit is costt, d the upper limit is. d d.,, the the fuctio F f t dt Proof: Use the defiitio of derivtive. df F h F lim d h h So, df d lim h h f t dt f t dt h lim f t dt. Note tht h h h h lim h h f t dt h f t dt h is the verge vlue of f from to + h. So, the Me h Vlue Theorem, there must eist some vlue c etwee d + h such tht f c pproches zero, c must pproch : lim f c f h. Therefore, df d f t dt h. As h h F lim h F f. h Note: A simplified w of thikig out the Fudmetl Theorem is tht ou re tkig the derivtive of tiderivtive. Sice these re iverses, the result is the origil fuctio. d E: Fid 7t dt d. Becuse the lower limit of itegrtio is costt, d the upper limit of itegrtio is, we c use the d Fudmetl Theorem. So d 7t dt 7. Pge 4 of Perso Pretice Hll 7 Clculus: Grphicl, Numericl, Algeric These otes re liged to the tetook refereced ove d to the College Bord Clculus AB curriculum.

15 AP Clculus Notes: Uit 6 Defiite Itegrls Specil Cse #: Wht if the upper limit of itegrtio is vrile epressio other th? We must use the chi rule. E: Fid d t dt. d if cos d B the Fudmetl Theorem d the chi rule, d 6 Solutio: cos d d cos t dt cos Derivtive of Specil Cse #: Wht if the vrile is the lower limit of itegrtio? We must use the properties of itegrtio to switch the limits of itegrtio. d E: Fid dt d. t Usig the properties of itegrls, dt dt t t. So d dt d. t Specil Cse #: Wht if there re vriles i oth the lower d upper limits of itegrtio? Use the properties of itegrtio to split them ito two itegrls. d for l t t dt We will choose costt,, tht is i the domi of f t l t t. (Note: Your choice is ritrr, ut must e E4: Fid d. i the domi of the fuctio.) For this emple, use =. Split the itegrl ito sum of two itegrls: d d l t tdt = l t tdt l t tdt So, l t tdt d = l t tdt l t tdt d To use the Fudmetl Theorem, we must switch the limits of itegrtio i the first itegrl, d use the chi rule i the secod itegrl. d tl tdt tltdt l l 4l d The Fudmetl Theorem of Clculus (Prt ) lso kow s the Itegrl Evlutio Theorem: If f is cotiuous t ever poit of,, d if F is tiderivtive of f o,, the f d F F Pge 5 of Perso Pretice Hll 7 Clculus: Grphicl, Numericl, Algeric These otes re liged to the tetook refereced ove d to the College Bord Clculus AB curriculum.

16 AP Clculus Notes: Uit 6 Defiite Itegrls Derivtio: B defiitio d the first Fudmetl Theorem, f t dt F C We kow tht f t dt FC, so it follows tht C F If we let, we hve f t dt F C. Sustitutig C. F, we hve f t dt FF E5: Evlute the itegrl. Fid the tiderivtive, d F of f. F We will use the followig ottio for F F :. 9 d Fidig Totl Are: To fid the re etwee the grph of.. Prtitio, with the zeros of f.. Itegrte f over ech suitervl.. Add the solute vlues of the itegrls. E6: Fid the re of the regio etwee the curve 9 Prtitios:,,,5. d d f d the -is over the itervl,, 9 d the -is o the itervl,5. Check our swer o the clcultor. Asolute vlue (s) c e foud i the Mth meu. Pge 6 of Perso Pretice Hll 7 Clculus: Grphicl, Numericl, Algeric These otes re liged to the tetook refereced ove d to the College Bord Clculus AB curriculum.

17 AP Clculus Notes: Uit 6 Defiite Itegrls Busiess Applictio: Ivetor E7: Suppose wholesler receives shipmet of cses of cd ever ds. The cd is sold to retilers t sted rte, d ds fter the shipmet rrives, the ivetor still o hd is I 5. Fid the verge dil ivetor. The fid the verge dil holdig cost if holdig o to cse costs cets d. Averge Dil Ivetor: vi Id 5d 5 55 cses Averge dil holdig cost: 55. $6.5 / d You Tr: Evlute the itegrl sec td. QOD: Epli the Fudmetl Theorem of Clculus (Prt ) i our ow words. Smple AP Clculus AB Em Questio(s):. Let f e differetile fuctio with f d f g f where? (A) (B) 5 (C) 65 (D) 67 (E) 6 5, d let g e the fuctio defied. Which of the followig is equtio of the lie tget to the grph of g t the poit g si t dt for. Let g e the fuctio give itervls is g decresig? (A) (B).77 (C).5.7 (D) (E).8. O which of the followig Pge 7 of Perso Pretice Hll 7 Clculus: Grphicl, Numericl, Algeric These otes re liged to the tetook refereced ove d to the College Bord Clculus AB curriculum.

18 AP Clculus Notes: Uit 6 Defiite Itegrls f. The grph of f, the derivtive of f, is the lie show i the figure ove. If f 5, the f (A) (B) (C) 6 (D) 8 (E) d d 4. si t dt 6 (A) cos (B) si 6 (C) si (D) si 6 (E) si Pge 8 of Perso Pretice Hll 7 Clculus: Grphicl, Numericl, Algeric These otes re liged to the tetook refereced ove d to the College Bord Clculus AB curriculum.

19 AP Clculus Notes: Uit 6 Defiite Itegrls Sllus Ojective:.6 The studet will pproimte defiite itegrl usig the trpezoidl sum. Recll: Are of Trpezoid Formul: A h E: Use 4 trpezoids of equl heights to pproimte the re uder the curve,. The fid the ect vlue. o the itervl Drw the grph d sketch the trpezoids. (Note: Trpezoid I is specil trpezoid with oe se equl to.) IV. I. II. III. Fid the res of ech trpezoid usig the formul give ove. Note: The ses of the trpezoids re the fuctio vlues for ech vlue of. 5 I. A II. A III. A IV. A Sum of the res: Note tht ll of the trpezoids hve the sme height. Also, trpezoids I & II, II & III, d III & IV shre commo se. So isted of fidig ech re idividull, we could put them ll together: A 4 Ect vlue: 8 d The trpezoidl pproimtio ws slightl lrger. Did ou epect it to e? Trpezoidl Rule: To pproimte f d, T... h, where h. Techer Note: Ecourge studets NOT to tr to memorize this formul. Usig the re of trpezoid formul with commolit of ses will llow them to recrete the rule. Also, the itervls m ot e costt, so the heights of the trpezoids m differ! Pge 9 of Perso Pretice Hll 7 Clculus: Grphicl, Numericl, Algeric These otes re liged to the tetook refereced ove d to the College Bord Clculus AB curriculum.

20 AP Clculus Notes: Uit 6 Defiite Itegrls E: The tle ws creted recordig the temperture ever hour from oo util midight. Use the trpezoidl rule to pproimte the verge temperture for the -hour period. Averge Temperture = f d You Tr: The tle elow shows the velocit of cr t secod time itervls. Estimte how fr the cr trveled usig trpezoidl sum. Compre this swer to the swers ou otied i the You Tr prolem few ds go (LRAM d RRAM). C ou drw coclusio sed o this compriso? Time Velocit (sec) (ft/sec) QOD: Wht is the reltioship etwee LRAM, RRAM, d Trpezoidl Sums. Prove this reltioship lgericll. Smple AP Clculus AB Em Questio(s): If trpezoidl sum overpproimtes 4 f d, d right Riem sum uderpproimtes f which of the followig could e the grph of (A) (B) (C) Time Noo Midight Temperture f? 4 d, (D) (E) Pge of Perso Pretice Hll 7 Clculus: Grphicl, Numericl, Algeric These otes re liged to the tetook refereced ove d to the College Bord Clculus AB curriculum.

Approximate Integration

Approximate Integration Study Sheet (7.7) Approimte Itegrtio I this sectio, we will ler: How to fid pproimte vlues of defiite itegrls. There re two situtios i which it is impossile to fid the ect vlue of defiite itegrl. Situtio: