00020002010016Definite Integrals.ACT00050200121_Area_dx_f(x).EAC010000000bcbÉ 1_Area_dx_f(x).EAC†Definite Integrals.ACT†0†3†=ˆ@†DŠ Ž Ž$ ^E Ž'$[ˆ4Using”X to †ld ˆ‡...Ž'††Ÿ 4Bounded by a curve, í¸-axis, a†8two vertical lines.\ŒkMore Details--->툌†íŽÍÈ$MThe key to understanding howˆset up an integral for fiŒ&area is this: ŽU Œ]bAŽa squ†+†cit of measure! Justˆ=nkˆt†expression as one-†„d dxˆˆ( other.p[†uˆË ˆm=ˆaŒÛ2[Œä%ˆäŠ©†za ver†ïhin rec†îgle is:ˆH‰ f(í¸)îdí¸î¨R†$-or-[Ž5 Q Š<-gˆBd ¶JE+The îS just sums these tiny rectangles fromˆ lower to upp† limit.[ŽJArea bounded by í¹=f(í¸)ˆ]! í¸-axis and vertical lines, í¸=aŠ í¸=b is:[Œ]ŠBŒiaŒqbŒyí¸Œ‚ (note:unitŒ–2Œž)ž§ î~Example 1[ŒÃ!Find‰ ažÄŠÁ=í¸ W,˜ÖŠÃ1Ã2žr î~Solution†ÙŽu†¿ ¬1œ¼’á=˜83ŽŒ|í¸=2Ž-Œ"í¸œ,š41Ž4=’12¢0ŽK-U1¢T\ˆšView the Area ->̈±†µÊˆ»¬N(ÜFinaFormÌ$N†Graph2D†ð†& 3ˆ†3 ˆLISTSYSˆ†@4ˆ< Modify ˆA$ˆ<STATCALC hˆdˆŒ< p\ˆx SequenceˆŒ,ˆxSheetˆOø|’ŠŒˆŒolveEqˆˆ´†€bwrôˆ´(Up†ˆ¥ŒtupFLG1 (Š<†Lis†{4D‰ˆPic†xÜViewWindœˆŒ_osve†b‡;¸xy†^‰ ÄH†ä‡6ŒÜ 䊗‰@è´ä’’ÈäŽP ‰  ˆ-Zˆ ˆ ’’’’$’’0’ˆ2Œ<’’H’’T’’`’’l’’x’’„’’† †œ†!†¨†"†´†#† À’‰Œ̆ %†؆&†ä†'†ð†(†ü†L!)‘¤¸†ä*††  †+Žl,’ˆ †T-† 8†.†D†0†P†1†\†2†h†3†t†4†€†5†Œ†E†˜†F†¤†H†°†I†¼†J†ȆK†Ô†L†à†M†ì†N†ø†¸O‘’‰6‹§‡Q‹¸†Á††¼†Ð R††ˆíSŽT†@†]‡I L†U ˆ^ŽÆ_’ˆF†æ`† X†aŽÖb’ˆ2†”† d’Œ•†p† ͆|†Άˆ†Ð’ˆZ†(׆  z¬†Ù†¸†Ú’=Ä’Û’ˆn‹| Financial‡–Format †ˆ † †ˆ‹­†2†systemä]list’^]=’_Œ’`Šaˆ $b~ŽhÁŽŽž¾ þ@þ€þÀÿ§@Á† ÿ@ÿx‡¾ˆˆx 9ˆŽ š!øÀŒ¾† a‡ seq_h†§b† NewFolde† ’ šÍ†0’ŠÌ ” —X††‡ î<ˆxª<Š'Žê‰ˆ‰ÁŠŒ‘,ö<þxþ´þðÿ,ÿh—¤ ‰`†ð‡F†¼†† †\ Œ ’,† †0†@†@PSheet1úê¬^Š2ž3ž 4†5ž ŒH3DÊ|èš|š†††Š˜| “i™” ™‰G‰K† 'Š Ž C^E ‰eŽÀ‰`†$«d«yŽ “¬  ˆB†ŒŒ†††œŽ †B  3xx^2Œ?Ž`– PlÆ$´$’³– p”x” –<”  ™€–`’ œ0™(1…0qy`ˆü#Y‡uYƒ‡= ˜–˜œšHª„Æ$ÄHp†‹ ˆ ’Ž †ež¢”1– àHàxà¨øØ¬ð†ü †h’e5‰yˆ” – œ™8Wˆâ[ˆê)1Tap Analysis/G-Solve/îSdx (type 1 then 2ŒOK)Ž9A Your turn...[ î~Example 2[Š#7Find the area bounded by f(í¸)=sinˆ,ˆ"í¸-axis,† =Œ?îŒH6ŽQ a†Q¨"2Rˆ†ŒxŒŒ’[Œ•\ˆ³Ex 2 Answer...툆"튈ՎJ A†À=îP(3)/2”JŽRYour turn again†NŽl—3[!¿í¸ÕG-4£#0‘ 2[Øõ Ex 3 QuestionŒô톆 €†Why is the area negative? Ž" Œ* KHint: DrawŠ.curv†4nd†< ink aboutŠH height ofŠrectangle (y-value).†[ŒeAct0200172_Area_dx_f(x)-g(x).EAC010000001d0cÉ 2_Area_dx_f(x)-g†.EAC†Definite Integrals.ACT†5†8†BˆE†IŠ Ž Œ ^E Ž'-[ˆ4Using”X to †ld ˆŒ...Ž'††¤-Bounded by two curves a†1ˆvertical linesˆ=ŠŠ[Œm ˆá=u†ÊŒO2ŽX7The a†ÿ of aˆH†`hin rectangle can be thoughtŠ(s:ˆZ‡<‰;=Hei† îWidth î ¨k =(Upper-Lower)Ž1Lˆ!Œñ f(í¸‰†îdí¸.7*where dí¸ is a very small change in í¸-values[\Œ To see th†< idea...Έ#†'Έ-Žˆˆ>Awup† `T’SDx€Š eˆ@"t7†  “2†"pŠ ŽDˆzvŠIŒ`r† ŽmA† 'LŽ'ˆ®C˜ u˜x˜yŽ †qˆqˆ¹ÄH ŽuŠÄ@ “ŠE@‡ Ð6–°†’Š#ˆ¾Qt“S)† ™Ž±Œ Š ‰ ÈÀ ŒU–Bx’˜‚Y™ ƒGa!6†„ˆB–! ƒ7sˆŽòÅ’… C@Ð6–°†P† x”Rsi ™Š†(Œ Š ††7Ç@" ††Žˆ"ˆ†a"Æ@ ”* †s ÉÀ …Œ–‚k `AUWt† Y‡†ˆ†&ˆšŽ‚ †‚Ç@ Š‚FšÉŒ/† U$y2Š ŽÕ †üĘ›ˆYˆŸˆéœˆo rehˆš›”³Š` ‡PÊÀ!•0.2î’(íëx-1)^2+1.9†‡x"Řb‡Š ÈÀ Õ P%ˆ† ˆ Ä@"ƒ ˆD@†"Ð6–°†)Ž/Š&8UB†vŽ2 ™††CÄ@ ŠCY†¢ˆ=…àlˆ‰AUWt†”Y‡kŒ` †  † ††µ1ÅH–F ††ËÈÀ–ˆ9†¢††0”0–èx’˜‚†" Y™ ƒGa!6†„ˆÜ–! ƒ7sˆŽÎˆìˆÚ†d1¢Úˆ”ˆ‚ˆ «ˆ†£0‡—Æ@!…B™…$g"R0†¿&xdD‚†Œ Š †Ä@  †Š A@Ð6–°ˆ"2EI`ƒ„yY™ZÐ$’Y ŽOŽˆ!†5Š!˜ˆŠŽ%ˆ«Ž-ˆE ŽÃ†vŽÎ[ŽŠ.c**The îS just sums these tiny rectangles fromˆ lower í¸-value limit toˆ7 upp¢. sArea bounded by í¹=f(í¸)ˆ gˆ , where Š is below Š%,and vertical lines, í¸=a ˆí¸=b†1 :[ (Š[-ŠX)ŒaŒbŒ$í¸Œ- [Œ6Ž î~Example 1[ŒS Find the a¢Éìñ=Ž'†VŒ„2Ž2ŽB +2Œ®¹ìò= †Ž/ fromˆÓ1Ü3ž  î~Solution\ˆñ1st I graph each>Ј3†7ʉtS Graph2D|† †Š3І ˆLISTSYSˆœ4N†Modify†; І($ˆ< STATCALC ô†Uˆ(ˆŒ< ü\ˆ< SequenceX†<,ˆxSheet†Š„†|’ŠŒ†¤ŽolveEq†ˆ´Lwr€†Í (UpˆŒ’ tupFLG1˜†<(Š<†Lis†{À†dD‰ˆPic†Ü ViewWind(†‰,x3б‡@‰EHˆ4’ˆÜŠy†rŽl† ˆ(y2’ˆ ˆ<äŠçЍÈä’´ÜäŽx À †ä† † Ì’’Ø’’ä’’ð’’ü’d’xˆnŒŒ ’ ,’´8’ÈD’ÜP’ð\’ˆn‹h’!‘t’"‘,€’#‘@Œ’$‘T˜’%‘h¤’&‘|°’'‘¼’(‘¤ȸ†ä)†Ô†  †*’à’+’ì’,’ø’-P’.d’0x’1Œ(’2 4’3´@’ˆŠËL’5ÜX’Eðd’F‘p’H‘|’I‘,ˆ’J‘@”’K‘T ’ˆ–‹k¬’M‘|¸’N‘Ä’O‘¤Ð’P‹¸܆ †äQ††è’R’ô’S(†@ŽT< ’]P†i ˆ^d’_x ’`Œ$’a (’b´,’”È0’Œ•Ü<’ÍðH’ΑˆÒŒБˆ–‹,††ˆŒ!†” system‰list’‡]=’‡‰’†ü‰’†ø‰’†ô~«ÁŽŽž¾ þ@þ€þÀÿÿ@ÿxÁŽŽž¾ þ@þ€þÀÿÿ@ÿx‡¸ÁŽŽž¾ þ@þ€þÀÿÿ@ÿx‡¸ÁŠŽž¾ þ@”†† a†– seq_histb† NewFoldeŠ­ˆ3 ’ systemä]l†0’^]=” –膆î<ˆxö<þxžðsystemä^]=ÁŠŒ’,]listö<ö<þxþðÃ,††† †RX’ ’,† †0†@†@PSheet1úê¬^Š2ž3úê¬^Sheet4˜5žŒ( 3D†† †0†@†PŠ,1è”<2ž3ž †|š†|š˜| Šˆ i™” ™†† Š Ž ˆ8 ø^„œ­ˆ½ŽÀ ††$’S’\¢œ“ ²<‡Žh†(¦d‰G 6y‡5¢‡)ŠòˆÀ 3x0.5*x^(2)+2† † 3x((x)/(2))’`– P8Æ$´$’– p”x” –<”  ™@”¨ ¬0’ †ó(1…0qy`ˆÌ#Y‡uYƒ‡  ˜–˜œšHª„Æ$ÎHš©š¶ª†Œ˜ ’ – ²1²J²c²|²•²®²Ç¢à˜Ø¬ð`–ü ™†S‹%[‹]We see from the graph† at í¹ìñ is aboveˆ1ò for all í¸-values between 1 and 3. So, we have:ŽeR‰˜Œví¸Œ€2Ž2ˆ +2-† í¸Œ 2Œ1Œ3’!R019 [ŒB Your turn...Ž î~Example 2[Œl" Find the area bounded by í¹ìñ=í¸‹Ž1 a†1†ò=ޝ’†¸/ª+2j , from í¸9-Ø]Œ]’"’ù2ŽÙ\#Hint: graph Each >Ј"†&ÊŠ,ÀS Graph2D,† †Š3Š@† ˆLISTSYSˆL4N†Modify†; €†($ˆ< STATCALC ¤†Uˆ(ˆŒ< ¬\ˆ< Sequence†<,ˆxSheet†Š4†|’ŠŒ°’olveEq†´ˆ´Lwr0†Í (Upˆ<’ tupFLG1H†<(Š<†Lis†{p†dD‰ˆPic†´Ü ViewWind؆‰,äб‡@ôxäŠ ŠŒä(  ä<´äP$Èäd’ÜäŽx < †ä† † H’’T’’`’’l’’x’’„’’’’œ’’¨’ ’´’!’À’"’Ì’#’Ø’$’ä’%’ð’&’ü’'‘@’(‘T’)‘hˆ¾Œ*‘|,’+‘8’ˆ‹§D¸†ä-†P†  †.’\’0’h’1’t’2’€’3’Œ’4’˜’5’¤’E’°’F’¼’H’È’I’Ô’J’à’K’ì’L’ø’M‘,’N‘@’O‘T’‰^‹k(’Q‘|‰ŒR‘@’S‘¤ˆ–ŒT‹¸X† †ä]††d† ˆ^’h’_’l’`’p’a’t’b’x’”’|’Œ•’ˆ’Í’ˆ2ŒÎ’ ’Ð’¬Ü††ˆŒ!†” system‰list’‡]=’‡‰’†ü‰’†ø‰’†ô~[ÁŽŽž¾ ¤@ÁŽŽž¾ þ@þ€þÀÿÿ@ÿxÁŽŽž¾ þ@þ€þÀÿÿ@ÿx‡¸ÁŽŽž¾ þ@þ€þÀÿÿ@ÿx‡¸ÁŠŽž° †† a†F seq_histb† NewFoldeŠ]ˆ3 ’ systemä]l†0’^]=” –˜††î<ˆxö<þxìðÒ´ emÁŠŒsystˆä]list’^]= ö<þxþ´<† ††† †’ ’,† †0†@†@PSheet1úê¬^Š2ž3ž 4ž 5ž@ r-value:í²|PSheet1èê¬^Š2ž3ž 4ž 5ž ŒH3D Šˆ i™” ™†† Š Ž ˆ8ø^„œ­† ††$’S’\¢œ”D²<‡Žh†(žŽ `– PÝÆ$´$“$– p”x” ,ˆˆ  ™€ `” ˜$’.ˆ>(1…0qy`ˆH#Y‡uYƒ†U ˜–0– šH®Æ$ÊHš©š¶´¨5°´þaþ ‰¼†Œ˜ °˜’ ®`–$ ™†OŠq[Œ<ŒŒ’\Œ Ex 2 Answer툆펴8Gˆ+’ The correct aŠ6 is: Area=0.9172258793ÿ[Œ€_-_-†Š– ŽYour turn again...Ž9 î~Example 3[ŒÍ$Find t††area bounded by í¹ìñ=(í¸-1)Œú2†«dˆ!ò=-í¸+3Ž™Fint: For the limits, findІ ersection po†- s of í¹ìñ a†$† ò.][\Œ Graph Each...̈ †!ÊŠ'ÀS Graph2D,† †Š3Š@† ˆLISTSYSˆL4N†Modify†; €†($ˆ< STATCALC ¤†Uˆ(ˆŒ< ¬\ˆ< Sequence†<,ˆxSheet†Š4†|’ŠŒ°’olveEq†´ˆ´Lwr0†Í (Upˆ<’ tupFLG1H†<(Š<†Lis†{p†dD‰ˆPic†´Ü ViewWind؆‰,äб‡@ôxäŠ ŠŒä(  ä<´äP$Èäd’ÜäŽx < †ä† † H’’T’’`’’l’’x’’„’’’’œ’’¨’ ’´’!’À’"’Ì’#’Ø’$’ä’%’ð’&’ü’'‘@’(‘T’)‘hˆ¾Œ*‘|,’+‘8’ˆ‹§D¸†ä-†P†  †.’\’0’h’1’t’2’€’3’Œ’4’˜’5’¤’E’°’F’¼’H’È’I’Ô’J’à’K’ì’L’ø’M‘,’N‘@’O‘T’‰^‹k(’Q‘|‰ŒR‘@’S‘¤ˆ–ŒT‹¸X† †ä]††d† ˆ^’h’_’l’`’p’a’t’b’x’”’|’Œ•’ˆ’Í’ˆ2ŒÎ’ ’Ð’¬Ü††ˆŒ!†” system‰list’‡]=’‡‰’†ü‰’†ø‰’†ô~[ÁŽŽž¾ ¤@ÁŽŽž¾ þ@þ€þÀÿÿ@ÿxÁŽŽž¾ þ@þ€þÀÿÿ@ÿx‡¸ÁŽŽž¾ þ@þ€þÀÿÿ@ÿx‡¸ÁŠŽž° †† a†F seq_histb† NewFoldeŠ]ˆ3 ’ systemä]l†0’^]=” –˜††î<ˆxö<þxìðÒ´ emÁŠŒsystˆä]list’^]= ö<þxþ´<† ††† †’ ’,† †0†@†@PSheet1úê¬^Š2ž3ž 4ž 5ž ‡›ŠX3D´|PSheet1èê¬^Š2ž3ž 4ž 5ž ŒH3D Šˆ i™” ™†† Š Ž ˆ8ø^„œ­† ††$’S’\¢œ”D²<‡Žh†(žŽ `– PÝÆ$´$“$– p”x” ,ˆˆ  ™` `’˜$’ˆ>(1…0qy†6 #Y‡uYƒ†U ˜–0– šH°`Æ$ÊH°xšÁšÎ¬°´âIâz«†Œ˜ °˜’ ®`–$ ™†OŠq[Œ<ŒŒ’[Ž\Œ% Ex 3 Answeríˆ!†í޼Gˆ’ÊArea=9/2 ÿ’OeAct0200123_Area_dy_f(y).EAC010000001dd6É 3_Area_dy_f(y).EAC†Definite Integrals.ACT†0†3†=ˆ@†DŠ Ž Ž$ ^E Ž'&[ˆ4Using”X to †ld ˆ‡...Ž'††Ÿ 6Bounded by a curve, í¹-axis, a†8two horizontal lines.\ŒmMore Details--->툎†’íµ%dThe key to understanding howˆset up an integral for fiŒ&area boˆ;d b†G†Oy-axis is still: Žl Œt"Aˆ5†"a squ†B†zit of measure!”2ާ GJust thinkˆ(ˆfexpression, f(í¹), asˆ€width†±d dyŽ height.Žid[†Z‰ ˆ=ˆ2[%‰Šà†ša ver‡&†‚ rec‡%gle is:ˆH‰I Œ‡îdí¹î¨R†$-or-[‘~ ˆ ŠÃ-gˆÉ› ¶JE+The îS just sums these tiny rectangles fromˆ lower to upp† limit.\Viewž;Έ†Î†$Š †ˆˆ5™™™™™™† `bG˜qQŠ Œ˜† ŽXŠ ŽDˆqvŠIŽQr† Ž^A† 'LŽ'ˆ¥C˜ u˜x˜yŽ †qˆqˆ "Ä@  ŽuŠÐÈÀ"Ž@‡Ð6–°Ž‡Ž%ŒªŠñˆÉ’`¢˜4ŽËÄ@!’p†Ä@#ƒŠ‚Hˆq Ð6–°††Š Œ ˜†† †0Ä@  І ˆB†ŠG@†RYˆ3Š9 ˜˜Y<Ž ˆ&ˆ* †K1ÄH#”]† †£ÈÀ!”s(íëx-0.85)^(1/2)+1.6†9Ž2ˆ“ †L"†L’L †HÈÀ^–¨˜Œô‡‰ g353pY™ žÄ£ƒ‹+F™b`‡yœe ™˜޳‰{œY’YE@†Ð6–°†ˆ† ™g353pˆ ˆˆŽ †††B1Ä@# ††XÈÀ ’-(íëx-0.85)^(1/2)+1.6†9Ž3††M"ÄH"”M†I †IŒ–©™™™™† †Œ´ˆ" ‘@"$`ˆ†TÄ@ Š¡C˜ì VG66‰Y …”7’ëÐD‰ œCD›/Š/†’z†&††–††CÄ@—ˆ˜ˆÞ¢ˆo†øÉHŽA’Ú”òY††‡½" Ä@  ††  ÈÀ!…Œ@†Ð6–°†™™™™† †02esda Y††KÄ@"ŠTA–C Qi–R6”7`H †SOŠŒC!ŽCB–†˜+”z˜CŠ †CšÚˆ˜ˆÞ¢ˆoˆŠ ˜Ú”òY†††††¿‰11ÄH ’F††9ŽdŠ‚†!‡[1Ç p*ˆ ŽžˆˆFˆ%Žˆ!ˆOˆˆ%‡D† ††† ††††††#Ž [ŽŠ>JArea bounded by í¸=f(í¹) the í¹-axis and vertical lines, í¹=cŠ í¹=d is:[Œ]ŠBŒicŒqdŒyí¹Œ‚ (note:unitŒ–2Œž)ާ î~Example 1†`Œ¼FindŠ až½Šº=Œjí¹-1 Zˆ¹ŠÕŠÂ1Â2.‘\‰" Graph Each ->̈"†!ÊUN0%FinaFormô$N†Graph2D 3Š, ˆLISTSYSˆ8†@4ˆ< Modify lˆPˆ<STATCALC ˆdˆŒ< ˜\ˆx S:equenceˆŒ,ˆxSheetˆO |’ŠŒœ’ olveEqˆ´†€`wrˆ´(Upˆ(’tupFLG14(Š<†Lis†{\D‰ˆPic† ÜViewWindĈŒ_osve†v‡ àxx†^‰‡(ìˆ(H†y2‰hЇ(ˆ´Šä<0ÈäŽP <‰) †ä† † ‡j9H’’T’’`’’l’’x’’„’’’’‡’¨’’´’’À’’̆†؆†ä† †ð†!†ü†$"‘@† #‘T‰6Œ$‘hˆZ† %‘|,†&‘8† !'‘¤D¸†ä(†P†  †)†W\†[*†h†+†t†ˆE€†-†Œ†.†˜†0†¤†1†°†2†¼†3†Ȇ4†Ô†5†à†E†ì†F†ø†H‘,† I†È† J‘T† K†Ç‰r†ÇL‘|ˆª† M†È@†N‘¤ˆ2†îO‹¸†ÉX††Ä†Ø P††d†eQ†ip†R†|†S†ˆ†T†”†]‡h  †} ˆ^ŽÎ_†¨†`†¬†aŽÞb†´’ˆ‚Šß¸’Œ•†Ć͆ІΆÜ’ˆ ‹/è’׆ô†Ø‘T‡lŽÙ‘h †Ú‘|† 6Û‘$‘¤ Financial‡¾Format †ˆ †Œ†+ˆ!– systemä]list’^]=’_Œ’`Šaˆ "b~ŽuÁŽŽž¾ þ@þ€þÀÕÁ†œ ÿÿ@ÿx‡¾ˆ¨Ž ž!ˈµ†† a‡/ seq_h†§b† NewFoldeF† ’ šÍ†0’ŠÌ ” —€†††MÐ<ˆ‰ Áˆˆ †µ ö<ö<þxþðÿ,ÿh§¤'’2-ŽVЇ^†††† †Œ&Œ,’ ’,† †0†!@†@PSheet1úê¬^Š2ž3ž 4†5ž@ r-value:íÈ|èš|š†††ŠŒÄ3DŠû “i™” ™‰G‰K† "Š Ž i^E ‰eŽÀ‰`†$«d‹Ÿ†·Œ” †L †† ®"†È’$†(œ†AŠ  †j 6y(y-3)^2”/ˆk 3x1¦2ˆkŽ`– PÀÆ$´$“– p”x” –<”  ™€–`’ œ0™r(1…0qy`‰(#LY‡uYƒ‡‘ ˜–˜œšH»†† – Œ ®Æ$p `š0’ ’¤˜°âIâzâ«Úܱ”™8 †f’e5‰y‰P”   `” †ˆ †Yˆ [ˆ î~Solution[Œí¹-1Œ#2Œ+1œí¹ŒD=ž#3Œ_-0=¢Ž{ƒ Your turn...Ž— î~Example 2[ŽLFind the area bounded by f(í¹)†…Žu†™»3Œî,ˆ?†Üaxis,† =1 a†Uí¹=2.\‰ Graph EachЈ©#†­Ê‰BN,%FinaFormô$N†Graph2D 3Š, ˆLISTSYSˆ8†@4ˆ< Modify lˆPˆ<STATCALC ˆdˆŒ< ˜\ˆx S:equenceˆŒ,ˆxSheetˆO |’ŠŒœ’ olveEqˆ´†€`wrˆ´(Upˆ(’tupFLG14(Š<†Lis†{\D‰ˆPic† ÜViewWindĈŒ_osve†v‡!àxx†^‰ìH†y2‰hЇ (‹|ˆ(ä<Š ÈäŽP 8‰+ †ä† † ‡lND’’P’’\’’h’’t’’€’’Œ’’˜’’¤’’°’†¼’†Ȇ†Ô††à† †ì†!†ø†("‘@† #‘T† $‘hˆªŒ%‘|(†&‘4† !'‘¤@¸†ä(†L†  †)†VX†Z*†d†+†p†,†|†-†ˆ†.†”†0† †1†¬†2†¸†3†ĆˆtІ5†܆E†è†F†ô†H‘,‰5†¬I†È †J†È† K‘h$’‰r‹ˆú†%M†È<†N‘¤ˆ‚†äO‹¸†ÉT††Ä†Ø P††`†eQ†il†R†x†S† „’ˆZŠS† ]‡l œ†} ˆ^ŽÏ_† ¤’ˆ–Š£¨† aŽàb†°Žü´’Œ•†À† ͆̆ΆØ’Ð’ä†׆ ð’ˆ2‹Wü†Ù‘h† Ú‘|† 5Û‘ ‘¤ Financia‰‘Format †ˆ †Œ†&ˆ!– systemä]ä^’† ˆ4’_ä`–†äaŒbˆ( ŠwLISTŽŽž¾ þ@þ€þÀш†Œî8þpþ°þðÿ0ÿp¡°Œœ7 ÿÿ@ÿxŽ ž!ˈ¸‡%† a†b‡1GrapŽph2D‰C‰G†? ” œÌˆ¼ ˜ —€† ††DÐ<†„Ž£ ˆ †« 0 ö<ö<þx†½,ÿh§¤ .Œ ‡J”ˆŒÒ‡d[†È†£† †Œ2Œ8’ ’,† †0†!@†@PSheet1@ÐDÐ=„Š2ž3ž 4†5††¼8†À7Ê|Œ ýˆËŽ|œ˜%$@ r-value:íˆû “i™” ™‰G‰K† Š Ž u^E ‰eŽÀ‰`†$«d‹«†Œ” †† ®"’$†(œ†AŠ ˆ  6yy^3/3ˆg 3x1¦2ˆgŽ`– P¼ÆN$´$“– p”x” –<”  ™€–`’ œ0™n(1…0qy`‰$#Y‡uYƒ‡ ˜–˜œšH“·Œ®–ˆH ®Æ$’’¢Ä#ˆ°þaþ þߞجð`—˜ †b’e5‰yˆ” Š ˆ `ˆŒ T ˆ Y [Š ŒŒ’[Ž\ˆ&Ex 2 Answer...íˆ$†"íŠNŒR Area=5/4 Žb’NŽVYour turn again†RŽp î~Example 3[ŽO Find the a†X bounded by f(í¹)=Žgí¹ŒÒ3ŒÚ,ˆ5í¹-axis,† =-3 a†Lí¹=0.Û Graph EachЈ%†ÊüN4%FinaFormô$N†Graph2D 3Š, ˆLISTSYSˆ8†@4ˆ< Modify lˆPˆ<STATCALC ˆdˆŒ< ˜\ˆx S:equenceˆŒ,ˆxSheetˆO |’ŠŒœ’ olveEqˆ´†€`wrˆ´(Upˆ(’tupFLG14(Š<†Lis†{\D‰ˆPic† ÜViewWindĈŒ_osve†v‡ àxx†^‰‡(ìˆ(H†y2‰hЇ(ˆ´ˆ(ä<†  ŠÈäŽP @‰, †ä† † ‡m!L’’X’’d’’p’’†ô,’ˆ’’”’’ ’’¬’’¸’’Ä’†І†܆†è† †ô† !‘,‹!Œ"‘@ †#‘Tˆæ†$‘h†  † %‘|0†&‘<† !'‘¤H¸†ä(†T†  †)†_`†c*†l†+†x†,†„†-††.†œ†ˆ]¨†1†´†2†À†3†̆4†؆5†ä†E†ð†F†ü†H‘,† I†È† J‘TˆæK†Ç‰"†ÇL†Ç8†M†ÇD† N†ÇP‡'O‹¸†Ç\††Â†Ö P††h†cQ†gt†R†€†S†Œ†T†˜†]‡j ¤†} ˆ^ŽÌ_†¬°†aŽÜb†¸†”†¼’Œ•†Ȇ ͆Ô†Άà‡»Ð’ì†׆ø† Ø‘T† Ù‘h† Ú‘|† 6Û‘(‘¤ Financial‡¾Format †ˆ †Œ†&ˆ!– systemä]ä^’† ˆ4’_ä`–†äaŒbˆ( ŠwLISTŽŽž¾ þ@þ€þÀш†Œî8þpþ°þðÿ0ÿp¡°Œœ7 ÿÿ@ÿxŽ ž!ˈ¸‡%† a†b‡1GrapŽph2D‰C‰G†? ” œÌˆ¼ ˜ —€† ††DÐ<†„Ž£ ˆ †« 0 ö<ö<þx†½,ÿh§¤ .Œ ‡J”ˆŒÒ‡d[†È†£† †Œ2Œ8’ ’,† †0†!@†@PSheet1@ÐDÐ=„Š2ž3ž 4†5††¼8†À7Ê|Œ ýˆËŽ|œ˜%$@ r-value:íˆû “i™” ™‰G‰K† Š Ž u^E ‰eŽÀ‰`†$«d‹«†Œ” †† ®"’$†(œ†AŠ  †j 6yy^(1/3)”/ˆk 3x0”-3ŽoŽ`– PPÄÆ$´$“ – p”x” –<”  ™€–`’ œ0™v(1…0qy`‰,#Y‡uYƒ‡• ˜–˜œšH‡%ޱˆ<'Ž – Š ®Æ$’’¢Ä#ˆ°þaþ þߞجð`—˜ †^’e5‰yˆ †¤ˆ$Y \† Ex 3 Question1íˆ&†"íŠ(|ˆ,Why is the area negative? Ž" Œ* JHint: DrawŠ.curv†4nd†< ink aboutŠHwidth ofŠUrectangle (x-value).¬ª2Œª'šª`Œ© WWhat woulˆqŽ­be if we changeŒlimit†Òo y=-3Š¢y=3? Tž§graph!‡8[ eAct0200174_Area_dy_f(y)-g(y).EAC010000001ec5É 4_Area_dy_f(y)-g†.EAC†Definite Integrals.ACT†5†8†BˆE†IŠ Ž Œ ^E Ž',[ˆ4Using”X to †ld ˆŒ...Ž'††¤/Bounded by two curves a†1ˆhorizontal linesˆ?ŠŒ[Œo ˆã=u†ÌŒQ2ŽZ7The a‡ of a ver†bhin rectangle can be thoughtŠ(s:ˆZ‡>‰==Len†îHeiˆ#î¨k =(Rˆ"-Left)0Lˆ!Œó f(í¹‰†îdí¹.?*where dí¹ is a very small change in í¹-values\To see th†4 idea...Έ†Îˆ%Žˆˆ6Awup† `T’SDx€Š eˆ@"t7†  “2†"pŠ ŽDˆrvˆŽ`r† Žm A† 'LŽ'ˆ¦C˜ u˜x˜yŽ †qˆqˆ¹"Ä@  ŽuŠÈÀ@‡Ð6–°Ž‡Ž%ŒªŒ 'wSVˆí ˜4†‰.†p#”pˆÝÆp†e `Š˜’ Œ†7Ä@" †Ä@ ŠM@†YÐ6–°†`ªe”UЊ † ŒY ’Y ŒY#ŽYL˜Y¨¾˜ŽYˆ] ¢]ˆ¶ ‡ÈÀ ŽÌ–¹ VW$&™&'Y™‘IGTRgˆ ƒt5Ž`ˆF †\1¢¹†\†\”Ï-(-5î’(íëx-3.1))^(1/2)+5.65†!9Ž9£‰„‡a"‹sK‰a Ð6–°†† 'wSVˆ Œ Š ††0Ä@ 3ˆBÄ@"ŠJ@†RY&`šY˜ŽYŠYH"’Yˆ€ˆy†1šˆs†9†ˆbÈÀ ”8(-5î’(íëx-3.1))^(1/2)+5.5Ž7ŽÄ”Q‰ÈÀ"ƒŠE'˜Å"6B•54ˆˆ’“™ W"§†D ÈÀ#—•ˆXY™#4D2”uˆ –b@RgŽNކ‹Cœ†U•¤`† ÐDŠ ††Ä@  '††’Ž"ˆŒ*"”*ˆÈÀ ŠcF'@†kÐ6–°†r…”xƒ¤„†D ÈÀ"ŒV–B!x`%x– — ™™aFwY™™H4!0ƒŽNˆÆˆ¨¢ ˆˆ`|Šˆ"ˆ¢ôˆý–ÊDœÊ3˜Bƒ‘‡‰Bp“™ W†Ñ£N!ŒÊ ¨ÊYux•&ˆ¾ ˜ ™f%r% hŠÊt2ˆ1–#AŽN‰zˆ="Œ !‡”††!†Š"Ž#† ÈÀ `ˆ)A'@†1Ð6–°†8E’$2a8˜p(#ˆFŠ*ŠWŒ Š ˆ† ††††ŽxŽiˆ)ˆ‰ˆ)ˆ}ˆ5ˆ1†††žAˆ¯† ††† †ŽÙˆ]Œå[ŽŠ.c**The îS just sums these tiny rectangles fromˆ lower í¹-value limit toˆ7 upp¢.žsnArea bounded by í¸=f(í¹)ˆ gˆ , where Š  is to the left of †.¸),and horizontal lines, í¹=c ˆí¹=d†< :[ (Šf-Šc)ŒcŒdŒ$í¹Œ- [Œ6Ž î~Example 1[ŒSFindŠ›a¢Ôìñ=Ž'†VŒ„2Ž2ŽB +2Œ®¸ìò= †Ž/ fromˆÓ1Ü3.ž¡ î~Solution\ˆò1st I graph each>Ј3$†7ʉlS Graph2D|† †Š3І ˆLISTSYSˆœ4N†Modify†; І($ˆ< STATCALC ô†Uˆ(ˆŒ< ü\ˆ< SequenceX†<,ˆxSheet†Š„†|’ŠŒ†¤ŽolveEq†ˆ´Lwr€†Í (UpˆŒ’ tupFLG1˜†<(Š<†Lis†{À†dD‰ˆPic†Ü ViewWind(†‰,x†Jг‰EHˆ2’`†ˆy3’x†ˆ(y4’ˆ ŠäŠçŠ Èä’¬ÜäŽx ¸ †ä† † Ä’’Ð’’Ü’’è’’ô’d†wŽx ’Œˆ2Œ $’´0’È<’ÜH’ðT’ ‘`’!‘l’"‘,x’#‘@„’ˆª‹W’%‘hœ’&‘|¨’'‘´’(‘¤À¸†ä)†̆  †*’Ø’+’ä’,’ð’-’ü’.d’0x’1Œ ’2 ˆnŒ3´8’4ÈD’5ÜP’Eð\’F‘h’H‘t’I‘,€’J‘@Œ’K‘T˜’L‘h¤’M‘|°’N‘¼’O‘¤È’ˆÒÔ† †äQ††à’R’ì’S’ø’T<’]P†i ˆ^d’_x’`Œ’a  ’b´$’”È(’Œ•Ü4’Íð@’ΑL’БX‘,ˆé†íˆŒ!†” system‰list’‡]=’‡‰’†ü‰’†ø‰’†ô~«ÁŽŽž¾ þ@þ€þÀÿÿ@ÿxÁŽŽž¾ þ@þ€þÀÿÿ@ÿx‡¸ÁŽŽž¾ þ@þ€þÀÿÿ@ÿx‡¸ÁŠŽž¾ þ@”†† a†– seq_histb† NewFoldeŠ­ˆ3 ’ systemä]l†0’^]=” –膆î<ˆxö<þxžðsystemä^]=ÁŠŒ’,]listö<ö<þxþðÃ,††† †RX’ ’,† †0†@†@ PSheet10ÐDĸ‰ŽŠ2ž30ÐDĸSheet40И5žr-value:í†9†† †0†@†PŠ< 1èê¬^ŒL2ž3ž 4ž 5ž ¤3Dˆ¯ ˆµˆ¹i™” ™†ˆŠ Ž ˆç ø^„œ­ˆ½ŽÀ ††$’S’\¢œ“ ²<‡Žh†(¦d ‡€  6yy^2/2+‡+‡f’/–‰{ 3x1††  3x3`– P/Æ$´$”T– p”x” –<”  ™@”¨ ¬0 ˆé(1…0qy`ˆÌ#Y‡uYƒ‡ ˜–˜œšHª„Æ$ÎHš©š¶´‡½†Œˆ’ – ¢)¢:¢K¢\¢m¢~¢¢ ¢±¢Â¢Ó¢Ø¬ð`–ü ™†S‹[‹]We see from the graph† at í¹ìñ is aboveˆ1ò for all í¸-values between 1 and 3. So, we have:ŽeR‰Œví¹Œ€2Ž2Ž+2- í¹ˆ 2Ž1Ž3Œ†!RŒ19Ž03[Œ9 Your turn...Ž î~Example 2[Œb# Find the area bounded by í¸ìñ=í¹œ‹ a†1†ò=Ž€2ŽN†¹ 0+Ž +1, from í¹=:-Úœ_¹=1.\Hint: graph Each >Љ%‡+ʉ1tS Graph2D|† †Š3І ˆLISTSYSˆœ4N†Modify†; І($ˆ< STATCALC ô†Uˆ(ˆŒ< ü\ˆ< SequenceX†<,ˆxSheet†Š„†|’ŠŒ†¤ŽolveEq†ˆ´Lwr€†Í (UpˆŒ’ tupFLG1˜†<(Š<†Lis†{À†dD‰ˆPic†Ü ViewWind(†‰,x†Jг‰EHˆ2’\† ˆy3’‰hŠ(y4’”†ˆ<äŠçЍÈä’´ÜäŽx À †ä† † Ì’’Ø’’ä’’ð’’ü’d’xˆnŒŒ ’ ,’´8’ÈD’ÜP’ð\’ˆn‹h’!‘t’"‘,€’#‘@Œ’$‘T˜’%‘h¤’&‘|°’'‘¼’(‘¤ȸ†ä)†Ô†  †*’à’+’ì’,’ø’-P’.d’0x’1Œ(’2 4’3´@’ˆŠËL’5ÜX’Eðd’F‘p’H‘|’I‘,ˆ’J‘@”’K‘T ’ˆ–‹k¬’M‘|¸’N‘Ä’O‘¤Ð’P‹¸܆ †äQ††è’R’ô’S(†@ŽT< ’]P†i ˆ^d’_x ’`Œ$’a (’b´,’”È0’Œ•Ü<’ÍðH’ΑˆÒŒБˆ–‹,††ˆŒ!†” system‰list’‡]=’‡‰’†ü‰’†ø‰’†ô~«ÁŽŽž¾ þ@þ€þÀÿÿ@ÿxÁŽŽž¾ þ@þ€þÀÿÿ@ÿx‡¸ÁŽŽž¾ þ@þ€þÀÿÿ@ÿx‡¸ÁŠŽž¾ þ@”†† a†– seq_histb† NewFoldeŠ­ˆ3 ’ systemä]l†0’^]=” –膆î<ˆxö<þxžðsystemä^]=ÁŠŒ’,]listö<ö<þxþðÃ,††† †RX’ ’,† †0†@†@PSheet1úê¬^Š2ž3úê¬^Sheet4˜5žr-value:톆 †0†@†PŠ<1è”L2ž3ž †|š†|šŒ¤3D Šˆ i™” ™†† Š Ž Š8ø^„œ­ˆ½ŽÀ ††$’S’\¢œ“ ²<‡Žh†(¦d‡J  6yy^‡'”2/(†+2)+‡W ˆ^ 3 x-1/2††  3ˆ1’`– P0Æ$´$’w– p”x” –<”  ™€–`’ œ0’ †ë(1…0qy`ˆÌ#Y‡uYƒ‡ ˜–˜œšHª„Æ$ÎHš©š¶ª†Œ˜ ’ – ²1²J²c²|²•²®²Ç¢à˜Ø¬ð`–ü ™†S‹%[ŒŒ’[Ž\Œ% Ex 2 Answeríˆ&†íp8ŒOThe correct aŠ1 is: ŒmArea=2.476021718–kˆsYour turn again...[† î~Example 3[Œ'$Find the area bounded by í¸ìñ=(í¹-1)ŒA2ŒI a†5†!ò=-í¹+3 Ža I[Hint: Fo††S limits, fd†N values ofŠupointŠintersection.]޲\ˆ» Graph Each†ÏЈ'†!ÊŠ'S Graph2DT† †Š3Šh† ˆLISTSYSˆt4N†Modify†; ¨†($ˆ< STATCALC ̆Uˆ(ˆŒ< Ô\ˆ< Sequence0†<,ˆxShee†]†Œ\†|’ŠŒØ’olveEq†´ˆ´LwrX†Í (Upˆd’ tupFLG1p†<(Š<†Lis†Ø˜†dD‰ˆPic†ìÜÜViewWin†L‡0‰,x†Jг‡EHˆ2’8†ˆä’P äŠGŠˆðŠ´ä’‹|ˆÈä’‡|ŒÜäŽx € †ä† † Œ’’˜’’¤’’°’’¼’’È’’Ô’’à’’ì’’ø’È’ Ü’!ðˆnŒ"‘(’#‘4’$‘,@’%‘@L’&‘TX’'‘hd’ˆn‹p’)‘|’*‘¤ˆ¸†ä+†”†  †,’ ’-’¬’.’¸’0’Ä’1’Ð’2’Ü’3’è’4’ô’5´ˆ½ŒEÈ ’FÜ’Hð$’I‘ˆ¾ŒJ‘<’K‘,ˆFŒL‘@T’M‘T`’N‘hl’O‘|x’P‘„’Q‘¤’R‹¸œ† †äS††¨’T’´’]’À†A ˆ^’Ä’_’È’`’Ì’a’Ð’b’Ô’”’Ø’Œ•’ä’Í’ð’Î’ü’ˆnŠó‘††ˆŒ!†” system‰ˆø’’‡ˆð’†üˆì’†øˆè’†ô( ‹…LIS‹zŽž” LISTŽŽž¾ þ@þ€þÀÿÿ@ÿxLISTLŽŽž¾ þ@þ€þÀÿÿ@ÿx‡¸STLIŽŽž¾ þ@þ€þÀÿÿ@ÿx‡¸TLISŽŽž¾ ¼@†e† a†b†qGrapŽph2Dˆƒˆ‡ ” systemä]ä^–† ˜ –À† †î<ˆö<þxìðsystemä]ä^’† † ’†,LISTˆ4Œö<þxþ´äð† ††† †"(’ ’,† †0†@†@PSheet1@ÐDÐ ?äŠ2ž3ž 4ž 5–@heetr-value:톆 †0†@†PSˆ,1ÈŒ û8† Š2ž3ž 4ž 5ž@˜| ˆˆ i™” ™†† Š Ž ˆ7 ø^„œ­ˆ½ŽÀ ††$’S’\¢œ“ ²<’h†(¦d ˆ˜ 6y(y-1)^2’Ü”-y+3ŠOŽ`– P‘¬$P† `” †$” – p”0” –<”  ™€–`’ œ0 ˆ–(1…0qy`ˆ„#Y‡uYƒ†­ ˜–˜œšHª„Æ$æH— šÀšÍ®°´ÎI†Œ˜ °à1ða’ ®`–$ ™†QŠÍ[Œ<ŒŒ’[Ž\Œ% Ex 3 Answeríˆ(†í,ŒP Area=9/2 Œ` ŽŽ˜_eAct02000f5_Area_Quiz.EAC010000001ea7É 5_Area_Quiz.EAC†Definite Integrals.ACT†-†0†:ˆ=†AŠ Ž Œ ^E Ž'"[ˆ4Using”X to †ld ˆ„...Ž'††œ!Practice what you have learned!Ž'ˆ/ˆÌ î~Example 1[ŒC,Fˆ_the a†ç bounded by í¸ìñ=í¹+1 a††ò=Ž6í¹Œ„2ŒŒ-1œ  from í¹=0ŒB¹=1ˆÁŠ’\Œð Graph each†ÞЈ†!ÊO(N\FinaForm$N†Graph2D†, 3Š@ ˆLISTSYSˆL4ˆ< Modify €ˆPˆ<STATCALC ¤ˆdˆŒ< ¬\ˆx S:equenceˆŒ,ˆxSheetˆO4|’ŠŒ°’ olveEqˆ´†€`wr0ˆ´(Upˆ<’tupFLG1H(Š<†Lis†{pD‰ˆPic†´ÜViewWind؈Œ_osve†v‡=ôxx†^‰‰†i†x2’ˆ<†}†y3’ˆðŠˆúŒˆ ŠäŽP \‰( †ä† † ‡i1h’’t’’€’’Œ’’˜’’¤’’†ù ’¼’’È’’Ô’’à’’ì††ø† ‘† ‘†  ‘,ˆZŒ!‘@(†"‘T4† #‘h@† $‘|L† %‘X† &‘¤d¸†ä'†p†  †ˆH|†u)†ˆ†*†”†+† †,†¬†-†¸†.†Ć0†І1†܆2†è†3†ô†Í4ðˆù†¯5†Ë †‹E†Ë† F‘,$† H‘@†%ª†%I†Ë<†J‘h†2†K†ÌT†L†Ì`† M†Ìl‡*N‹¸†Ìx††Ç†Û O††„†€P†„†Q†œ†R†¨†S†´†ˆUÀ†]‡u ̆‘ ˆ^ŽÑ_†Ô†ˆq؆aŽáb†àä’Œ•†ð† ͆ü’Α,’ˆªŠ)’בT ˆ‹k,’Ù‘|8’Ú‘D’Û‘¤‰š "† FinancialFormat †ˆ †Œ†ˆ!– systemä]list’^]=’_Œ’`Šaˆ #b~ŽuÁŽŽž¾ þ@þ€þÀ­Áœ ÿÿ@ÿx‡¸Žž8ŽŠ–V¸øøˆ³†† a‡F seq_h†¥b†îNewFolde‹]ˆ3 ’ šË†0’ŠÊ ” —˜†††M¨<Š#ŽäˆþˆŒ› ŠŒ‘,ö<þxþ´þðÿ,ÿh—¤†È‡m‡p“;,]list’^]= ˆ0ÁŠ6††† †ŽTŽ[Ž ’,† †0†!@†@PSheet1úê¬^Š2ž3ž 4†5ž ä8ä7Ì|èš|š†††ŠCŒÄ3DŠû “i™” ™‡€‰ƒ‡‡‘ކ’^E ‰eŽÀ‰`†$¸†´Œ˜ ¤†M†† ²6’$†(œ†AŠ †~  6yy+1”+ˆ`Ž (y^2-1)/2ˆƒ 3x0¦1ˆƒŽ`– PìÆ$´$“3– p”x” –<”  ™@"”¨ ¬0™ž(1…0qy`‰@#Y‡uYƒ‡½ ˜†ñˆ²(– –®Æ$ÊH’ ’¤È%°´þaþ þߞؠð‰@ `Œ †  ™’e5‰yˆ” – œ”<Zˆˆ [ˆŒŒ’[Ž\Œ% Ex 1 Answer툆íŽN8ŒOThe correct aŠ1 is: Ž Œv Area=11/6ˆkŠ{Žs î~Example 2[ŽV&Find t†[area bounded by í¹ìñ=í¸cos(í¸ŒÓ2Ž ), í¸-axisˆ =0†’dˆ Œmî4.–ú‰ Hint: graph Each >Ј"ˆûÊ‘(N4%FinaFormô$N†Graph2D 3Š, ˆLISTSYSˆ8†@4ˆ< Modify lˆPˆ<STATCALC ˆdˆŒ< ˜\ˆx S:equenceˆŒ,ˆxSheetˆO |’ŠŒœ’ olveEqˆ´†€`wrˆ´(Upˆ(’tupFLG14(Š<†Lis†{\D‰ˆPic† ÜViewWindĈŒ_osve†v‡àxx2‰‰ì‰THˆ‡‡ ‰1ˆy††Œ*†|ˆ(ä<ˆ ŠÈäŽP @‰- †ä† † ‡n!L’’X’’d’’p’’†õ+’ˆ’’”’’ ’’¬’’¸’’Ä’†І†܆†è† †ô† !‘,‹!Œ"‘@ †#‘Tˆæ†$‘h†  † %‘|0†&‘<† !'‘¤H¸†ä(†T†  †)†_`†c*†l†+†x†,†„†-††.†œ†ˆ]¨†1†´†2†À†3†̆4†؆5†ä†E†ð†F†ü†H‘,† I†È† J‘TˆæK†Ç‰"†ÇL†Ç8†M†ÇD† N†ÇP‡'O‹¸†Ç\††Â†Ö P††h†cQ†gt†R†€†S†Œ†T†˜†]‡j ¤†} ˆ^ŽÌ_†¬°†aŽÜb†¸†”†¼’Œ•†Ȇ ͆Ô†Άà‡»Ð’ì†׆ø† Ø‘T† Ù‘h† Ú‘|† 6Û‘(‘¤ Financial‡¾Format †ˆ †Œ†+ˆ!– systemä]list’^]=’_Œ’`Šaˆ "b~ŽuÁŽŽž¾ þ@þ€þÀÕÁ†œ ÿÿ@ÿx‡¾ˆ¨Ž ž!ˈµ†† a‡/ seq_h†§b† NewFoldeF† ’ šÍ†0’ŠÌ ” —€†††MÐ<ˆ‰ Áˆˆ †µ ö<ö<þxþðÿ,ÿh§¤'’2-ŽVЇ^†††† †Œ&Œ,’ ’,† †0†!@†@PSheet1úê¬^Š2ž3ž 4†5ž ä8ä7Ì|èš|š†††ŠŒÄ3DŠû “i™” ™‰G‰K† #Š Ž ˆ†’^E ‰eŽÀ‰`†$«d‹Ÿ†¶Œ” †M †† ®"†ÇV’$†(œ†AŠ ˆO 6y0”ä/4’?ˆt 3xx*àP(x^2)ˆoŽ`– PÄÆ$´$“ † @ˆ™Tp”x” –<”  ™€–`’ œ0™v(1…0qy`‰,#Y‡uYƒ‡• ˜–˜œšH‰iжˆ>Ž – Š ®Æ$†V(T ˆT8#† `’"’+¢œ°âIâzâ«Úܘجð`—˜ †r’e5‰y‰D ’e5‰y`– †ˆ$Yˆ [ŠŒŒ’[Ž\Œ% Ex 2 AnsweríˆíŽM<ŒOThe correct aŠ1 is: ŒmArea=0.2892343946774–o†ˆ¬ î~Example 3[ŽYFind t†^area bounded by í¹ìñ=Ží¸ŒÛ2Œã-í¸-2¦ 3+8 E†²Œdí¸-axis between its poin† of inters†âion with˜4.4 GraphŠž...̈âÊ‘6¸N(°FinaForm¸$N†Graph2D†܆& 3Šð ˆLISTSYSˆü†@4ˆ< Modify 0ˆPˆ<STATCALC TˆdˆŒ< \\ˆx SequenceˆŒ,ˆxSheetˆOä|’ŠŒ`’o lveEqˆˆ´†~`wràˆ´(Upˆì’tupFLG1ø(Š<†Lis†{ D‰ˆPic†dÜViewWin†ˆˆŒ_osve†v‡4¤x䉉°Œä’¼ ä’È´ä’ÔÈäŽP à‰ †ä† † ‡^†— ’ø’†¬!’<’P’d(’†u4’ˆ2Š@’†rL’†rX’†rd’ Üp’!ð|† "‘ˆ† #‘”† $‘, † %‘@¬† &‘T¸† '‘hÄ’‰‹І)‘܆ 0*‘¤è¸†ä+†ô†  †,ˆŒ-( †„.<† 0P$† 1d†%†%2x<†3ŒH† 4 T† 5´`† EÈl† F†Ö x’ˆZŠó„†ŽI†Ö† J†Öœ† K†Ö¨† L†Ö´† M†ÖÀ† N†Ö̆ O‘|؆ P†Öä† Q†Öð‡0R‹¸†Öü††Ñ†å S††‡Oˆ˜†]‡[ †A ˆ^†ˆÖ_†(†ˆ´,†a†0†b†4†‰u8’Œ•†D† ͆P†Ά\†‰hh†ׇ¸t’؆€†Ù†Œ†Ú†˜†Û†K¤‘h Financial‡‚Forma†n †ˆ † †ˆŒE‡¥” systemä]listˆŠ^]=’_Œ†`Šaˆ " b~ÁŽŽž¾ þ@þ€þÀÿÏ@ÁŒ ÿ@ÿx‡¸°:ŒŽXŠ”XÖ¸Œ»† a†ö seq_h†­b† NewFolde‹ ˆ3 ’ ˆÔ’Ó†0’ŠÎ ” —H††‡î<ˆxÒ<ŒÄ ˆˆ †š<‡’“  ö<ö<þxþðÿ,ÿh†<N† ‰ †Œ †ˆo† †0†!@†0PSheet1úê¬^Š2ž3ž 4†5ž ŒH3DÊ|èš|š†††ŠL˜| “i™” ™†† Š Ž ˆ†’^E ‰9ŽÀ‰`†$«d«yŽ “€ « ‰µ†J†8 ††† yŠŽ † `– P2Æ$´$’y– p”x” –<”  ™`”`ª$˜ã (1…0qy`ˆä#Y‡uYƒˆH˜–˜œšH°`Æ$ÊHšðu#r2Šä™¤ˆ‹Š‚ Œ˜ ’ˆ]– Â9ÂZÂ{œ½¶Þ˜Ø¬ð†”ü ™’e5‰yˆ” – œ™8VˆÇ[‡ŒŒ’[Ž\Œ% Ex 3 Answerí‡N팒 `† The correct answer is: Ž Œ'Area=-0.5215710894387”&ž/†d[Ša††r î~Example 4[}Find t†‚area bounded by í¸-îŒ£í¹Ž­ =3,†+†Ž2Œ=2†¹=0†Ädˆ 1 \Œá Graph eachЈ†ŸÊŒ§ÀS Graph2D,† †Š3Š@† ˆLISTSYSˆL4N†Modify†; €†($ˆ< STATCALC ¤†Uˆ(ˆŒ< ¬\ˆ< Sequence†<,ˆxSheet†Š4†|’ŠŒ°’olveEq†´ˆ´Lwr0†Í (Upˆ<’ tupFLG1H†<(Š<†Lis†{p†dD‰ˆPic†´Ü ViewWind؆‰,äб‡@ôxäŠ ŠŒä(  ä<´äP$Èäd’ÜäŽx < †ä† † H’’T’’`’’l’’x’’„’’’’œ’’¨’ ’´’!’À’"’Ì’#’Ø’$’ä’%’ð’&’ü’'‘@’(‘T’)‘hˆ¾Œ*‘|,’+‘8’ˆ‹§D¸†ä-†P†  †.’\’0’h’1’t’2’€’3’Œ’4’˜’5’¤’E’°’F’¼’H’È’I’Ô’J’à’K’ì’L’ø’M‘,’N‘@’O‘T’‰^‹k(’Q‘|‰ŒR‘@’S‘¤ˆ–ŒT‹¸X† †ä]††d† ˆ^’h’_’l’`’p’a’t’b’x’”’|’Œ•’ˆ’Í’ˆ2ŒÎ’ ’Ð’¬Ü††ˆŒ!†” system‰ˆø’’‡ˆð’†üˆì’†øˆè’†ô( ‹]LISTŽŽž¾  @LISTŽŽž¾ þ@þ€þÀÿÿ@ÿxLISTLŽŽž¾ þ@þ€þÀÿÿ@ÿx‡¸STLIŽŽž¾ þ@þ€þÀÿÿ@ÿx‡¸TLISŽŽžª †=† a†b†IGrapŽph2Dˆ[ˆ_ ” systemä]ä^–† ˜ –˜† †î<ˆö<þxìðÒ´ emLIST† Œsystˆä]ä^’† ˆ2 ö<þxþ´<† ††† †’ ’,† †0†@†@ PSheet1<ÐDm°‰RŠ2ž3ž 4ž 5ž@ r-value:í²|PSheet1Œ ýŠ2ž3ž 4ž 5ž@r-value:í ˆˆ i™” ™ˆwŠ Ž ˆ7 ø^„œ­†šˆ†¡ ††$’S’\¢œ”D²<’h†(žŽ `– PÝÆ$´$“$– p”x” ,ˆˆ  ™€ `” ˜$’.ˆ>(1…0qy`ˆH#Y‡uYƒ†U ˜–0– šH®Æ$ÊHš©š¶´¨5°´þaþ ‰¼†Œ˜ °˜’ ®`–$ ™†OŠq[Œ<ŒŒ’[Ž\Œ% Ex 4 Answeríˆ †í޼@ŒOThe correct aŠ1 is: Ž Œv Area=3.05161†2”seAct010008main.ACT0001020012eActivity Save.EAC010000001ea7É 5_Area_Quiz.EAC†Definite Integrals.ACT†-†0†:ˆ=†AŠ Ž Œ ^E Ž'"[ˆ4Using”X to †ld ˆ„...Ž'††œ!Practice what you have learned!Ž'ˆ/ˆÌ î~Example 1[ŒC,Fˆ_the a†ç bounded by í¸ìñ=í¹+1 a††ò=Ž6í¹Œ„2ŒŒ-1œ  from í¹=0ŒB¹=1ˆÁŠ’\Œð Graph each†ÞЈ†!ÊO(N\FinaForm$N†Graph2D†, 3Š@ ˆLISTSYSˆL4ˆ< Modify €ˆPˆ<STATCALC ¤ˆdˆŒ< ¬\ˆx S:equenceˆŒ,ˆxSheetˆO4|’ŠŒ°’ olveEqˆ´†€`wr0ˆ´(Upˆ<’tupFLG1H(Š<†Lis†{pD‰ˆPic†´ÜViewWind؈Œ_osve†v‡=ôxx†^‰‰†i†x2’ˆ<†}†y3’ˆðŠˆúŒˆ ŠäŽP \‰( †ä† † ‡i1h’’t’’€’’Œ’’˜’’¤’’†ù ’¼’’È’’Ô’’à’’ì††ø† ‘† ‘†  ‘,ˆZŒ!‘@(†"‘T4† #‘h@† $‘|L† %‘X† &‘¤d¸†ä'†p†  †ˆH|†u)†ˆ†*†”†+† †,†¬†-†¸†.†Ć0†І1†܆2†è†3†ô†Í4ðˆù†¯5†Ë †‹E†Ë† F‘,$† H‘@†%ª†%I†Ë<†J‘h†2†K†ÌT†L†Ì`† M†Ìl‡*N‹¸†Ìx††Ç†Û O††„†€P†„†Q†œ†R†¨†S†´†ˆUÀ†]‡u ̆‘ ˆ^ŽÑ_†Ô†ˆq؆aŽáb†àä’Œ•†ð† ͆ü’Α,’ˆªŠ)’בT ˆ‹k,’Ù‘|8’Ú‘D’Û‘¤‰š "† FinancialFormat †ˆ †Œ†ˆ!– systemä]list’^]=’_Œ’`Šaˆ #b~ŽuÁŽŽž¾ þ@þ€þÀ­Áœ ÿÿ@ÿx‡¸Žž8ŽŠ–V¸øøˆ³†† a‡F seq_h†¥b†îNewFolde‹]ˆ3 ’ šË†0’ŠÊ ” —˜†††M¨<Š#ŽäˆþˆŒ› ŠŒ‘,ö<þxþ´þðÿ,ÿh—¤†È‡m‡p“;,]list’^]= ˆ0ÁŠ6††† †ŽTŽ[Ž ’,† †0†!@†@PSheet1úê¬^Š2ž3ž 4†5ž ä8ä7Ì|èš|š†††ŠCŒÄ3DŠû “i™” ™‡€‰ƒ‡‡‘ކ’^E ‰eŽÀ‰`†$¸†´Œ˜ ¤†M†† ²6’$†(œ†AŠ †~  6yy+1”+ˆ`Ž (y^2-1)/2ˆƒ 3x0¦1ˆƒŽ`– PìÆ$´$“3– p”x” –<”  ™@"”¨ ¬0™ž(1…0qy`‰@#Y‡uYƒ‡½ ˜†ñˆ²(– –®Æ$ÊH’ ’¤È%°´þaþ þߞؠð‰@ `Œ †  ™’e5‰yˆ” – œ”<Zˆˆ [ˆŒŒ’[Ž\Œ% Ex 1 Answer툆íŽN8ŒOThe correct aŠ1 is: Ž Œv Area=11/6ˆkŠ{Žs î~Example 2[ŽV&Find t†[area bounded by í¹ìñ=í¸cos(í¸ŒÓ2Ž ), í¸-axisˆ =0†’dˆ Œmî4.–ú‰ Hint: graph Each >Ј"ˆûÊ‘(N4%FinaFormô$N†Graph2D 3Š, ˆLISTSYSˆ8†@4ˆ< Modify lˆPˆ<STATCALC ˆdˆŒ< ˜\ˆx S:equenceˆŒ,ˆxSheetˆO |’ŠŒœ’ olveEqˆ´†€`wrˆ´(Upˆ(’tupFLG14(Š<†Lis†{\D‰ˆPic† ÜViewWindĈŒ_osve†v‡àxx2‰‰ì‰THˆ‡‡ ‰1ˆy††Œ*†|ˆ(ä<ˆ ŠÈäŽP @‰- †ä† † ‡n!L’’X’’d’’p’’†õ+’ˆ’’”’’ ’’¬’’¸’’Ä’†І†܆†è† †ô† !‘,‹!Œ"‘@ †#‘Tˆæ†$‘h†  † %‘|0†&‘<† !'‘¤H¸†ä(†T†  †)†_`†c*†l†+†x†,†„†-††.†œ†ˆ]¨†1†´†2†À†3†̆4†؆5†ä†E†ð†F†ü†H‘,† I†È† J‘TˆæK†Ç‰"†ÇL†Ç8†M†ÇD† N†ÇP‡'O‹¸†Ç\††Â†Ö P††h†cQ†gt†R†€†S†Œ†T†˜†]‡j ¤†} ˆ^ŽÌ_†¬°†aŽÜb†¸†”†¼’Œ•†Ȇ ͆Ô†Άà‡»Ð’ì†׆ø† Ø‘T† Ù‘h† Ú‘|† 6Û‘(‘¤ Financial‡¾Format †ˆ †Œ†+ˆ!– systemä]list’^]=’_Œ’`Šaˆ "b~ŽuÁŽŽž¾ þ@þ€þÀÕÁ†œ ÿÿ@ÿx‡¾ˆ¨Ž ž!ˈµ†† a‡/ seq_h†§b† NewFoldeF† ’ šÍ†0’ŠÌ ” —€†††MÐ<ˆ‰ Áˆˆ †µ ö<ö<þxþðÿ,ÿh§¤'’2-ŽVЇ^†††† †Œ&Œ,’ ’,† †0†!@†@PSheet1úê¬^Š2ž3ž 4†5ž ä8ä7Ì|èš|š†††ŠŒÄ3DŠû “i™” ™‰G‰K† #Š Ž ˆ†’^E ‰eŽÀ‰`†$«d‹Ÿ†¶Œ” †M †† ®"†ÇV’$†(œ†AŠ ˆO 6y0”ä/4’?ˆt 3xx*àP(x^2)ˆoŽ`– PÄÆ$´$“ † @ˆ™Tp”x” –<”  ™€–`’ œ0™v(1…0qy`‰,#Y‡uYƒ‡• ˜–˜œšH‰iжˆ>Ž – Š ®Æ$†V(T ˆT8#† `’"’+¢œ°âIâzâ«Úܘجð`—˜ †r’e5‰y‰D ’e5‰y`– †ˆ$Yˆ [ŠŒŒ’[Ž\Œ% Ex 2 AnsweríˆíŽM<ŒOThe correct aŠ1 is: ŒmArea=0.2892343946774–o†ˆ¬ î~Example 3[ŽYFind t†^area bounded by í¹ìñ=Ží¸ŒÛ2Œã-í¸-2¦ 3+8 E†²Œdí¸-axis between its poin† of inters†âion with˜4.4 GraphŠž...̈âÊ‘6¸N(°FinaForm¸$N†Graph2D†܆& 3Šð ˆLISTSYSˆü†@4ˆ< Modify 0ˆPˆ<STATCALC TˆdˆŒ< \\ˆx SequenceˆŒ,ˆxSheetˆOä|’ŠŒ`’o lveEqˆˆ´†~`wràˆ´(Upˆì’tupFLG1ø(Š<†Lis†{ D‰ˆPic†dÜViewWin†ˆˆŒ_osve†v‡4¤x䉉°Œä’¼ ä’È´ä’ÔÈäŽP à‰ †ä† † ‡^†— ’ø’†¬!’<’P’d(’†u4’ˆ2Š@’†rL’†rX’†rd’ Üp’!ð|† "‘ˆ† #‘”† $‘, † %‘@¬† &‘T¸† '‘hÄ’‰‹І)‘܆ 0*‘¤è¸†ä+†ô†  †,ˆŒ-( †„.<† 0P$† 1d†%†%2x<†3ŒH† 4 T† 5´`† EÈl† F†Ö x’ˆZŠó„†ŽI†Ö† J†Öœ† K†Ö¨† L†Ö´† M†ÖÀ† N†Ö̆ O‘|؆ P†Öä† Q†Öð‡0R‹¸†Öü††Ñ†å S††‡Oˆ˜†]‡[ †A ˆ^†ˆÖ_†(†ˆ´,†a†0†b†4†‰u8’Œ•†D† ͆P†Ά\†‰hh†ׇ¸t’؆€†Ù†Œ†Ú†˜†Û†K¤‘h Financial‡‚Forma†n †ˆ † †ˆŒE‡¥” systemä]listˆŠ^]=’_Œ†`Šaˆ " b~ÁŽŽž¾ þ@þ€þÀÿÏ@ÁŒ ÿ@ÿx‡¸°:ŒŽXŠ”XÖ¸Œ»† a†ö seq_h†­b† NewFolde‹ ˆ3 ’ ˆÔ’Ó†0’ŠÎ ” —H††‡î<ˆxÒ<ŒÄ ˆˆ †š<‡’“  ö<ö<þxþðÿ,ÿh†<N† ‰ †Œ †ˆo† †0†!@†0PSheet1úê¬^Š2ž3ž 4†5ž ŒH3DÊ|èš|š†††ŠL˜| “i™” ™†† Š Ž ˆ†’^E ‰9ŽÀ‰`†$«d«yŽ “€ « ‰µ†J†8 ††† yŠŽ † `– P2Æ$´$’y– p”x” –<”  ™`”`ª$˜ã (1…0qy`ˆä#Y‡uYƒˆH˜–˜œšH°`Æ$ÊHšðu#r2Šä™¤ˆ‹Š‚ Œ˜ ’ˆ]– Â9ÂZÂ{œ½¶Þ˜Ø¬ð†”ü ™’e5‰yˆ” – œ™8VˆÇ[‡ŒŒ’[Ž\Œ% Ex 3 Answerí‡N팒 `† The correct answer is: Ž Œ'Area=-0.5215710894387”&ž/†d[Ša††r î~Example 4[}Find t†‚area bounded by í¸-îŒ£í¹Ž­ =3,†+†Ž2Œ=2†¹=0†Ädˆ 1 \Œá Graph eachЈ†ŸÊŒ§ÀS Graph2D,† †Š3Š@† ˆLISTSYSˆL4N†Modify†; €†($ˆ< STATCALC ¤†Uˆ(ˆŒ< ¬\ˆ< Sequence†<,ˆxSheet†Š4†|’ŠŒ°’olveEq†´ˆ´Lwr0†Í (Upˆ<’ tupFLG1H†<(Š<†Lis†{p†dD‰ˆPic†´Ü ViewWind؆‰,äб‡@ôxäŠ ŠŒä(  ä<´äP$Èäd’ÜäŽx < †ä† † H’’T’’`’’l’’x’’„’’’’œ’’¨’ ’´’!’À’"’Ì’#’Ø’$’ä’%’ð’&’ü’'‘@’(‘T’)‘hˆ¾Œ*‘|,’+‘8’ˆ‹§D¸†ä-†P†  †.’\’0’h’1’t’2’€’3’Œ’4’˜’5’¤’E’°’F’¼’H’È’I’Ô’J’à’K’ì’L’ø’M‘,’N‘@’O‘T’‰^‹k(’Q‘|‰ŒR‘@’S‘¤ˆ–ŒT‹¸X† †ä]††d† ˆ^’h’_’l’`’p’a’t’b’x’”’|’Œ•’ˆ’Í’ˆ2ŒÎ’ ’Ð’¬Ü††ˆŒ!†” system‰ˆø’’‡ˆð’†üˆì’†øˆè’†ô( ‹]LISTŽŽž¾  @LISTŽŽž¾ þ@þ€þÀÿÿ@ÿxLISTLŽŽž¾ þ@þ€þÀÿÿ@ÿx‡¸STLIŽŽž¾ þ@þ€þÀÿÿ@ÿx‡¸TLISŽŽžª †=† a†b†IGrapŽph2Dˆ[ˆ_ ” systemä]ä^–† ˜ –˜† †î<ˆö<þxìðÒ´ emLIST† Œsystˆä]ä^’† ˆ2 ö<þxþ´<† ††† †’ ’,† †0†@†@ PSheet1<ÐDm°‰RŠ2ž3ž 4ž 5ž@ r-value:í²|PSheet1Œ ýŠ2ž3ž 4ž 5ž@r-value:í ˆˆ i™” ™ˆwŠ Ž ˆ7 ø^„œ­†šˆ†¡ ††$’S’\¢œ”D²<’h†(žŽ `– PÝÆ$´$“$– p”x” ,ˆˆ  ™€ `” ˜$’.ˆ>(1…0qy`ˆH#Y‡uYƒ†U ˜–0– šH®Æ$ÊHš©š¶´¨5°´þaþ ‰¼†Œ˜ °˜’ ®`–$ ™†OŠq[Œ<ŒŒ’[Ž\Œ% Ex 4 Answeríˆ †í޼@ŒOThe correct aŠ1 is: Ž Œv Area=3.05161†2”seAct0301da00200008000000001a9cxœíÝlçÇñ™õúÇš%¬í%1 M6ÄI–°$þ!­ïjŽbÀ§iz„xÁŽlLbzùnX;Ž»v]Ê¡ºãZ¡:ñG𣡖äø#UHŠ®(Z¢SîÄQ•k+rßgæÙggœ†äšôý’{æ»3»3»ë}>ûÌzÇ~;•²ßŽtÚïX Õ¬ZËõa€õ%µc׎î-–%÷ÁõˆeõõnÚžÞ¾KæíêòÖu˾ÙdYÞ|¥7ßèÎÏ»”¿¼Á?ß¶vÉâ¶æúâù¦ó‹ó÷æææ›óMùÆÀ|C`>°}ík 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