÷ƒ’À;è TeX output 1997.04.03:1643‹ÿÿÿÿ y ý£ ? ýä’§*¡óKñ`y cmr10²18.01–UUSolutions,“Problem“Set“5ŽŸ‘Mó !",š cmsy10¸ŽŽŽ‘WxŽ‘[q˲5.2‘UU#4.ŽŸqΑWCalculate‘UUó  b> cmmi10µd‘ª¨Ÿñæ^óú±u cmex10«ŽŸù<$‘ :Oµx–8ณ²2Ž‘ :OŸw‰feïûŸ (Öµx–8à²+“3ŽŽŽŽ‘"]}Ÿñæ^«ŽŽ‘)¹ñ².ŽŸqÏ‘Wóò"V cmbx10¹Solution:ŽŽŽŸ%OúŸô0 ‘frWµd‘ª¨Ÿñæ^«ŽŸù<$‘ :Oµx–8ณ²2Ž‘ :OŸw‰feïûŸ (Öµx–8à²+“3ŽŽŽŽ‘"]}Ÿñæ^«ŽŽŽ’Ÿ`º²=Ÿù<$‘úK(µx–8à²+“3)µd²(µx“¸“²2)“¸“²(µx“¸“²2)µd²(µx“²+“3)Ž‘úKŸw‰fe‘~&Ÿ (Ö‘7%M(µx–8à²+“3)ŸýróÙ“ Rcmr7±2ŽŽŽŽŽ’ ä„óý': cmti10º(quotient‘“çrule)ŽŽŽŸÃä’Ÿ`º²=Ÿù<$‘úK[(µx–8à²+“3)“¸“²(µx“¸“²2)]‘ª¨µdxŽ‘úKŸw‰fe[Ë€Ÿ (Ö‘Kú²(µx–8à²+“3)Ÿýr±2ŽŽŽŽŽ‘cÀ²=Ÿù<$‘5Ž‘úKŸw‰fe#3ŒŸ (Ö(µx–8à²+“3)Ÿýr±2ŽŽŽŽŽ‘* ²µdx:ŽŽŽŽŸ+AÁ‘M¸ŽŽŽ‘WxŽ‘[q˲5.2‘UU#9.ޤ‘WCalculate‘UUµd²(µxŸü^ÿ±2Ž‘|sŸø$û¸pŽ‘ ÑÊŸø$û‰feïüŸÛ²3µx–8à²+“2ŽŽŽ‘(ÁÆ).ŽŸ‘W¹Solution:ŽŽŽŸ)þ÷Ÿéâ ‘\µd²(µxŸü^ÿ±2Ž‘|sŸø$û¸pŽ‘ ÑÊŸø$û‰feïüŸÛ²3µx–8à²+“2ŽŽŽ‘(ÁÆ)Ž’¡tt=‘ǵxŸü^ÿ±2Ž‘|sµd²(3µx–8à²+“2)Ÿü^ÿ±1ó 0e—rcmmi7´=±2Ž‘ʪ²+“(3µx“²+“2)Ÿü^ÿ±1´=±2Ž‘ ‘ʵdxŸü^ÿ±2ŽŽ’•ðrº(pr–ÿ}'o“duct‘“çrule)ŽŽŽŸÏ?’¡tt²=‘ǵxŸü^ÿ±2Ž‘µS¸Ÿù<$‘l²1Ž‘lŸw‰feŸ (Ö2ŽŽŽŽ‘ ŸG(3µx–8à²+“2)ŸûÞÿó O!â…cmsy7·±1´=±2Ž‘Ñ˵d²(3µx“²+“2)“+“(3µx“²+“2)ŸûÞÿ±1´=±2Ž‘ ‘ʲ2µx‘ª¨dxŽ’•ðrº(p‘ÿ}'ower‘“çrule)ŽŽŽŸþ’¡tt²=Ÿù<$‘¹Û3µxŸü^ÿ±2Ž‘'µdxŽ‘úKŸw‰fe5HåŸ V½²2(3µx–8à²+“2)Ÿýr±1´=±2ŽŽŽŽŽ‘<¯C²+Ÿù<$‘l4µx²(3µx–8à²+“2)‘ª¨µdxŽ‘lŸw‰fe;mŸ V½‘ÝIJ2(3µx–8à²+“2)Ÿýr±1´=±2ŽŽŽŽŽ‘Bj˲=Ÿù<$‘ ø15µxŸü^ÿ±2Ž‘µS²+‘8à8µxŽ‘úKŸw‰fe5HåŸ V½²2(3µx–8à²+“2)Ÿýr±1´=±2ŽŽŽŽŽ‘Ÿ÷ºP‰fe ŸE°²83ŽŽŽ‘°X¸‘Dz3µ:²0183495º,“c˜omp˜ar˜e˜d“to“²163µ=²54–Ǹ“²3µ:²0185185º.Ž¡‘M¸ŽŽŽ‘WxŽ‘[q˲5.3–UUCompute“the“folloš¸ãwing“inde nite“in˜tegrals.ަ‘Wã#3.ŽŽŽ‘mŸ÷ñÄ«RŽ‘sª­²(µxŸü^ÿ±2Ž–µS²+›8àµxŸü^ÿ±3Ž“²+˜µxŸü^ÿ±4Ž‘|s²)‘ª¨µdx².ŽŸ‘m¹Solution:ŽŽŽŸœy’žÙeŸòc‡«ZŽ’¨Ùf²(µxŸûÞÿ±2Ž–µS²+›8àµxŸûÞÿ±3Ž“²+˜µxŸûÞÿ±4Ž‘|s²)‘ª¨µdx‘Dz=Ÿù<$‘úK1Ž‘úKŸw‰feŸ (Ö3ŽŽŽŽ‘ -µxŸûÞÿ±3Ž“²+Ÿù<$‘l1Ž‘lŸw‰feŸ (Ö4ŽŽŽŽ› ŸGµxŸûÞÿ±4Ž“²+Ÿù<$‘l1Ž‘lŸw‰feŸ (Ö5ŽŽŽŽ˜µxŸûÞÿ±5Ž“²+‘8àµC(ã:ŽŸÀS‘R㌲#14.ŽŽŽ‘mŸ÷ñÄ«RŽ‘uUUµxŸü^ÿ±2Ž‘|sŸøÌ¸pŽ‘ ÑÊŸøÌ‰fe·Ÿ3èµxŽŽŽ‘3Ždx².ŽŽŸŽŒ‹* y ý£ ? ýä‘m¹Solution:ŽŽŽŸté’¯ûÇŸòc‡«ZŽ’»¦pµxŸûÞÿ±2Ž‘|sŸøO¸pŽ‘ ÑÊŸøO‰fe·Ÿ°ãµxŽŽŽ‘3Ždx–Dz=“Ÿòc‡«ZŽ‘qÁµxŸûÞÿ±5´=±2Ž‘ ‘ʵdx“²=Ÿù<$‘úK2Ž‘úKŸw‰feŸ (Ö7ŽŽŽŽ‘ -µxŸûÞÿ±7´=±2Ž‘ʪ²+‘8àµC(ã:ŽŸ"¹Ô‘R㌲#22.ŽŽŽ‘mŸòc‡«ZŽ‘xª«Ÿñæ^ŽŸù<$’ƒÔ²6Ž’:RŸw‰fe 3Ÿ (ÖµxŸýr±4ŽŽŽŽŽ’ŽÙô²+–8à2µxŸûÞÿ±3´=±2Ž‘ʪ¸“²5Ÿñæ^«ŽŽ’Ì |µdx².ŽŸû•‘m¹Solution:ŽŽŽŸ"ËøŸõŒ‘{GbŸòc‡«ZŽ’†ò Ÿñæ^ŽŸù<$’’y²6Ž’²Ÿw‰fe 3Ÿ (ÖµxŸýr±4ŽŽŽŽŽ’!T²+–8à2µxŸûÞÿ±3´=±2Ž‘ʪ¸“²5Ÿñæ^«ŽŽ’ÚTܵdx–Dz=“Ÿòc‡«ZŽ‘ Dz(6µxŸûÞÿ·±4Ž‘ õT²+–8à2µxŸûÞÿ±3´=±2Ž‘ʪ¸“²5)‘ª¨µdxŽŽŸµ«’…€B²=‘Ç6–8ฟù<$‘O¢²1Ž‘lŸw‰fe ÇŸ (Ö¸²3ŽŽŽŽ‘feµxŸûÞÿ·±3Ž‘ õT²+“2“¸Ÿù<$‘l²2Ž‘lŸw‰feŸ (Ö5ŽŽŽŽ› ŸGµxŸûÞÿ±5´=±2Ž‘ʪ¸“²5µx“²+“µC‘~4²=Ÿù<$‘úK¸²2Ž‘úKŸw‰fe ÇŸ (Ö‘IȵxŸýr±3ŽŽŽŽŽ‘-}²+Ÿù<$‘l4Ž‘lŸw‰feŸ (Ö5ŽŽŽŽ˜µxŸûÞÿ±5´=±2Ž‘ʪ¸“²5µx“²+“µC(ã:ŽŽŽŽŸ/5¸‘R㌲#43.ŽŽŽ‘mŸòc‡«ZŽŸù<$‘yÝÞ²(1–8à+“ŸøÌ¸pŽ‘ Ž7ŸøÌ‰fe·Ÿ3èµxŽŽŽ‘ES²)Ÿü^ÿ±1´=±4ŽŽ‘yÝÞŸw‰fe3ž:Ÿ 3á‘ÈäŸøÌ¸pŽ‘;ŸøÌ‰fe·Ÿ3èµxŽŽŽŽŽŽ’°Yódx²,–UUusing“the“substitution“µu–Dz=“1–8à+“ŸøÌ¸pŽ‘ Ž7ŸøÌ‰fe·Ÿ3èµxŽŽŽ‘ES².Ž© /‘m¹Solution:ŽŽ’¢Ž*ºIf›“çµu–Dz=“1–8à+“ŸøÌ¸pŽ‘ Ž7ŸøÌ‰fe·Ÿ3èµxŽŽŽ‘ESº,˜then˜µdu‘Dz=Ÿù<$‘ …1Ž‘úKŸw‰fe tŸ 3á2ŸøÌ¸pŽ‘UWŸøÌ‰fe·Ÿ3èµxŽŽŽŽŽŽŽ‘äšdxº,˜soŽŸ,&ÞŸö£5’¥lŸòc‡«ZŽŸù<$’²Jk²2(1–8à+“ŸøÌ¸pŽ‘ Ž7ŸøÌ‰fe·Ÿ3èµxŽŽŽ‘ES²)Ÿü^ÿ±1´=±4ŽŽ’²JkŸw‰fe8ž;Ÿ 3á‘Èä²2ŸøÌ¸pŽ‘UWŸøÌ‰fe·Ÿ3èµxŽŽŽŽŽŽŽ’íÆdxŽ’²²=‘ÇŸòc‡«ZŽ‘qÁ²2µuŸûÞÿ±1´=±4Ž‘ï5PSfile=6.6.2.eps llx=0 lly=0 urx=182 ury=182 rwi=1080ŽŽ¡‘W¹Solution:ŽŽ’ŒŽ*ºThe–üÈline“is“the“gr›ÿ}'aph“of“µy‘"ñ²=‘ǸŸü ‘33±3Ž‘33Ÿ£&‰feüsŸ¿˜2ŽŽŽŽ‘bÙµx–ðZ²+“3º.‘g7We–üÈwant“the“ar˜e˜a“underަ‘Wthe›Kgr–ÿ}'aph,‘Y«b“etwe“en˜µx–Dz=“0˜ºand˜µx“²=“2˜º(the˜µxº-inter–ÿ}'c“ept).‘RThis˜is˜given˜by˜theަ‘Winte–ÿ}'gr“alŽŸ3‘nYßòc‡«ZŽŸôÜi‘xYı2ŽŸ@‘sçý0ŽŽ‘|Ö7²(¸Ÿù<$‘33²3Ž‘33Ÿw‰feŸ (Ö2ŽŽŽŽ›fgµx–8à²+“3)‘ª¨µdx‘Dz=‘úKŸñæ^«Ž‘ V¿¸Ÿù<$‘33²3Ž‘33Ÿw‰feŸ (Ö4ŽŽŽŽ˜µxŸûÞÿ±2Ž‘µS²+“3µxŸñæ^«ŽŽ‘CBŸñæ^ŽŽŸóøÛ‘HK`±2ŽŸ,‘HK`0ŽŽ‘OŽë²=›Ç¸Ÿù<$‘33²3Ž‘33Ÿw‰feŸ (Ö4ŽŽŽŽ‘ ŸG¸“²2ŸûÞÿ±2Ž‘µS²+“3“¸“²2“¸“²0˜=˜3µ:ŽŸ© ‘WºWe–F)ar›ÿ}'e“c˜omputing“the“ar˜e˜a“of“a“triangle“with“b˜ase“2“and“height“3.‘°\Theަ‘Wformula–“çµA‘Dz=Ÿü ‘úK±1Ž‘úKŸ£&‰feüsŸ¿˜2ŽŽŽŽ‘ )ñµbh“ºfr›ÿ}'om“elementary“ge˜ometry“gives“3“as“wel‘‚Øl.ŽŸ·R‘M¸ŽŽŽ‘WxŽ‘[q˲6.6‘UU#7.Ž¡‘WThe–Þicurvš¸ãe“µy‘f²=‘«µxŸü^ÿ±3Ž‘µ¸‘”B²9µx“²has“one“arc˜h“abGo˜v˜e“the“µx²-axis.‘ Find“the“area“ofަ‘Wthe–UUregion“under“the“arc¸ãh.ŽŸt¸ê‘>ï5PSfile=6.6.7.eps llx=0 lly=0 urx=382 ury=290 rwi=1422ŽŽ¡‘W¹Solution:ŽŽ’ŒŽ*ºThe–v›ar›ÿ}'ch“is“b˜etwe˜en“the“µxº-inter˜c˜epts“at“µx–Dz=“¸²3–v›ºand“µx–Dz=“0º,‘|wsoަ‘Wthe›“çar–ÿ}'e“a˜of˜the˜r“e“gion˜isŽŸ3‘qÒsŸòc‡«ZŽŸôÜi‘{Òt±0ŽŸ@‘w`­·±3ŽŽ’‚!²(µxŸûÞÿ±3Ž–µS¸‘8à²9µx²)‘ª¨µdx›Ç²=‘úKŸñæ^«ŽŸù<$‘ ‰ò²1Ž‘ ‰òŸw‰feŸ (Ö4ŽŽŽŽ‘½&µxŸûÞÿ±4Ž“¸Ÿù<$‘l²9Ž‘lŸw‰feŸ (Ö2ŽŽŽŽ‘ ŸGµxŸûÞÿ±2Ž‘|sŸñæ^«ŽŽ‘BýŸñæ^ŽŽŸóøÛ‘Gg±0ŽŸ,‘Gg·±3ŽŽ‘Tê§²=˜0–8ณ²(Ÿù<$‘3381Ž‘33Ÿw‰fe Ÿ (Ö‘€4ŽŽŽŽ–ŸH¸Ÿù<$‘l²81Ž‘lŸw‰fe Ÿ (Ö‘€2ŽŽŽŽ“)˜=Ÿù<$‘úK81Ž‘úKŸw‰fe Ÿ (Ö‘€4ŽŽŽŽ‘-€µ:ŽŽŸŽŒ‹)— y ý£ ? ýä‘M¸ŽŽŽ‘WxŽ‘[q˲6.6‘UU#15.ŽŸ'j‘WFind–Othe“area“bGounded“bš¸ãy“the“curv˜e“µy‘"ñ²=‘Ç2µx‘¬Ö²+Ÿù<$‘yÐ1Ž‘à Ÿw‰fe 3Ÿ (ÖµxŸýr±2ŽŽŽŽŽ‘F˲,‘Qthe“µx²-axis“and“the“linesŽŸÍБWµx–Dz=“1–UUand“µx–Dz=“3.ޤ‘W¹Solution:ŽŽ’ŒŽ*ºNote–˜>that“if“µx–ž·>“²0º,›ÙSthen‘˜>µy‘ú>“²0º,˜so–˜>the“r–ÿ}'e“gion–˜>is“ab‘ÿ}'ove“theŽŸ ‘Wµxº-axis.‘™–The›“çar–ÿ}'e“a˜isŽ©‰ý‘q§sŸòc‡«ZŽŸôÜi‘{§t±3ŽŸ@‘w5­1ŽŽ’Οñæ^«Ž’‰+²2µx‘8à²+Ÿù<$‘Ú1Ž‘lŸw‰fe 3Ÿ (ÖµxŸýr±2ŽŽŽŽŽ‘ÒÕŸñæ^«ŽŽ’·f·µdx–Dz=‘úKŸñæ^«Ž‘ V¿µxŸûÞÿ±2Ž‘µS¸Ÿù<$‘Ç¡²1Ž‘lŸw‰fe·Ÿ (ÖµxŽŽŽŽ‘ VbŸñæ^«ŽŽ‘1="Ÿñæ^ŽŽŸóøÛ‘6„@±3ŽŸ,‘6„@1ŽŽ‘=Ç˲=“(9–8ฟù<$‘l²1Ž‘lŸw‰feŸ (Ö3ŽŽŽŽ‘ ŸG)“¸“²(1“¸“²1)‘Ç=Ÿù<$‘úK26Ž‘úKŸw‰fe Ÿ (Ö‘€3ŽŽŽŽ‘-€µ:ŽŸñΑM¸ŽŽŽ‘WxŽ‘[q˲6.6–UU#“22.ŽŸ‘WIf–UUµn“²is“pGositiv¸ãe,“thenŽŸãr’Ãןòc‡«ZŽŸôÜi’Í×±1ŽŸ@’ÉeM·±1ŽŽ’ÕÌiµxŸûÞÿ´nŽ‘&µdx‘Dz=Ÿù<$‘%ìµxŸü^ÿ´n±+1ŽŽ‘-~Ÿw‰fe9yŸ (Öµn–8à²+“1ŽŽŽŽ‘š*Ÿñæ^«ŽŽŸó‘-‘"áH±1ŽŸnÚ‘"áH·±1ŽŽ‘/Hdµ:ŽŸÇ$‘W²Whš¸ãy–UUis“this“calculation“incorrect“if“µn“²is“a“negativ˜e“n˜um˜bGer“¸6²=‘Ǹ²1?Ž¡‘W¹Solution:ŽŽ’ŒŽ*ºThe–8$c›ÿ}'alculation“uses“the“F‘ÿ;¼undamental“The˜or˜em“of“Calculus,ŽŸñÏ‘Wwhich–È€applies“to“Ÿòc‡«ZŽŸôÜi‘È´bŽŸ@‘ VºaŽŽ‘wmµf›²(µx²)‘ª¨µdx“ºwhen“µf˜²(µx²)“ºis“c‘ÿ}'ontinuous“on“²[µa;‘ª¨b²]º.‘7`If“µn“ºisŽŸœy‘Wne‘ÿ}'gative,‘óthen›Ëzµf‘²(µx²)–Ç=“µxŸü^ÿ´nŽ‘<øºis˜not˜c‘ÿ}'ontinuous˜on˜the˜interval˜²[¸²1µ;‘ª¨²1]º.‘VÇIt˜is˜notŽŸ&o‘Weven–“çde ne›ÿ}'d“at“µx–Dz=“0º,–“çso“we“haven‘þúP't“even“de ne˜d“the“inte˜gr˜al“Ÿòc‡«ZŽŸôÜi‘ “è±1ŽŸ@‘ "!·±1ŽŽ‘‰=µxŸûÞÿ´nŽ‘&µdxº.ŽŸã–‘M¸ŽŽŽ‘WxŽ‘[q˲6.6–UUFind“the“v‘ÿqÇalue“of“eacš¸ãh“de nite“in˜tegral.ŽŸ˜6‘RãŒ#26.ŽŽŽ‘mŸòc‡«ZŽŸôÜi‘w±4ŽŸ@‘rŽ<1ŽŽ‘}'ŸøO¸pŽ’…|uŸøO‰fe·Ÿ°ãµxŽŽŽ’ŒÞ9dx².ޤ@‘m¹Solution:ŽŽŽ¦’ªµGŸòc‡«ZŽŸôÜi’´µH±4ŽŸ@’°C1ŽŽ’ºÜcµxŸûÞÿ±1´=±2Ž– ‘ʵdx›Ç²=Ÿù<$‘-~2Ž‘-~Ÿw‰feŸ (Ö3ŽŽŽŽ‘ `²µxŸûÞÿ±3´=±2Ž“Ÿñæ^«ŽŽŸóøÛ‘"ð¶±4ŽŸ,‘"ð¶1ŽŽ‘*4A²=Ÿù<$‘úK2Ž‘úKŸw‰feŸ (Ö3ŽŽŽŽ‘ -(8–8ณ²1)˜=Ÿù<$‘úK14Ž‘úKŸw‰fe Ÿ (Ö‘€3ŽŽŽŽ‘-€µ:ŽŸ&¦v‘R㌲#31.ŽŽŽ‘mŸòc‡«ZŽŸôÜi‘w±1ŽŸ@‘rŽ<0ŽŽ‘{|v²(µx–8ณµxŸûÞÿ±2Ž‘|s²)‘ª¨µdx².Ž¡‘m¹Solution:ŽŽŽ¦’—Y‘Ÿòc‡«ZŽŸôÜi’¡Y’±1ŽŸ@’œçË0ŽŽ’¥Ö²(µx–8ณµxŸûÞÿ±2Ž–|s²)‘ª¨µdx‘Dz=‘úKŸñæ^«ŽŸù<$‘ ‰ò²1Ž‘ ‰òŸw‰feŸ (Ö2ŽŽŽŽ‘½&µxŸûÞÿ±2Ž‘µS¸Ÿù<$‘l²1Ž‘lŸw‰feŸ (Ö3ŽŽŽŽ‘ ŸGµxŸûÞÿ±3Ž“Ÿñæ^«ŽŽ‘BýŸñæ^ŽŽŸóøÛ‘Gg±1ŽŸ,‘Gg0ŽŽ‘Nª¦²=Ÿù<$‘úK1Ž‘úKŸw‰feŸ (Ö2ŽŽŽŽ– f_¸Ÿù<$‘l²1Ž‘lŸw‰feŸ (Ö3ŽŽŽŽ“=Ÿù<$‘úK1Ž‘úKŸw‰feŸ (Ö6ŽŽŽŽ‘ -µ:ŽŸ'Š‘R㌲#38.ŽŽŽ‘mŸòc‡«ZŽŸôÜi‘w±2ŽŸ@‘rŽ<1ŽŽ‘}'Ÿñæ^«Ž’„ƒ’µx‘8à²+Ÿù<$‘Ç¡1Ž‘lŸw‰fe·Ÿ (ÖµxŽŽŽŽ‘ VbŸñæ^«ŽŽ’¥í‚ŸóøÛ±2Ž’­¿Eµdx².ŽŽŸŽŒ‹3½ y ý£ ? ýä‘m¹Solution:ŽŽŽŸ¸ðŸúþÆ‘rR1Ÿòc‡«ZŽŸôÜi‘|R2±2ŽŸ@‘wàk1ŽŽ’‚yMŸñæ^«Ž’‰ÕÁµx‘8à²+Ÿù<$‘Ç¡1Ž‘lŸw‰fe·Ÿ (ÖµxŽŽŽŽ‘ VbŸñæ^«ŽŽ’«?±ŸóøÛ±2Ž’³tµdxŽ’Çý²=‘ÇŸòc‡«ZŽŸôÜi‘ DZ2ŽŸ@‘UR1ŽŽ‘CŒ²(µxŸûÞÿ±2Ž‘µS²+–8à2“+“µxŸûÞÿ·±2Ž› ¼t²)‘ª¨µdx–Dz=“(µxŸûÞÿ±3Ž‘|sµ=²3–8à+“2µx“¸“µxŸûÞÿ·±1Ž˜²)]ŸûÞÿ±2ŽŸ™á1ŽŽŽŽŸòÔ’Çý²=‘Ç(Ÿü ‘33±8Ž‘33Ÿ£&‰feüsŸ¿˜3ŽŽŽŽ››¹²+–8à4“¸Ÿü ‘l±1Ž‘lŸ£&‰feüsŸ¿˜2ŽŽŽŽ˜²)“¸“²(Ÿü ‘33±1Ž‘33Ÿ£&‰feüsŸ¿˜3ŽŽŽŽ˜²+“2“¸“²1)‘Ç=Ÿü ‘úK±29Ž‘úKŸ£&‰feøæŸ¿˜‘þ:6ŽŽŽŽ‘ &dµ:ŽŽŽŽŸ+¿E‘R㌲#40.ŽŽŽ‘mŸòc‡«ZŽŸôÜi‘w´@L=±4ŽŸ@‘rŽ<´@L=±6ŽŽ’†-q²sinŽ’”7µt‘ª¨dt².ŽŸ¾‘m¹Solution:ŽŽŽŸ‘P‘mŸòc‡«ZŽŸôÜi‘w´@L=±4ŽŸ@‘rŽ<´@L=±6ŽŽ’†-q²sinŽ’”7µt›ª¨dt–Dz=“¸˜²cosŽ‘¸ßµt²]ŸúÎ:´@L=±4ŽŸ™”´@L=±6ŽŽ‘IÞ²=“cosŽ‘*§(µ•[Ù=²6)¸²cosŽ‘ c(µ“=²4)–Ç=“Ÿ÷=V¸pŽ‘ oŸ÷=V‰feŸª²3ŽŽŽ‘pµ=²2¸Ÿ÷=VpŽ‘UWŸ÷=V‰feŸª²2ŽŽŽ‘ UXµ=²2“=Ÿù<$‘úKŸ÷ºP¸pŽ‘ O¢Ÿ÷ºP‰feŸE°²3ŽŽŽ‘ˆƒ¸‘8àŸ÷ºPpŽ‘ Ž7Ÿ÷ºP‰feŸE°²2ŽŽŽŽ‘úKŸw‰fe&㎟ (Ö‘ñÇ2ŽŽŽŽ‘, µ:ŽŸž¸‘M¸ŽŽŽ‘WxŽ‘[q˲6.7‘UU#3.ޤ¢¿‘WFind–UUthe“area“bGounded“b¸ãy“µy[ÙŸü^ÿ±2Ž‘Ÿd²=›ÇµxŸü^ÿ±3Ž‘ÑȲand“µx˜²=˜4.ŽŸuäo‘>ï5PSfile=6.7.3.eps llx=0 lly=0 urx=470 ury=290 rwi=1750ŽŽ¡‘W¹Solution:ŽŽ’ŒŽ*ºThis–oßis“twic›ÿ}'e“the“ar˜e˜a“under“the“gr˜aph“of“µy‘±m²=‘U”µxŸü^ÿ±3´=±2Ž‘©ºb˜etwe˜enŽ© ‘Wµx–Dz=“0–“çºand“µx–Dz=“4º,–“çso“the“ar–ÿ}'e“a–“çis“given“byŽŸ©}’œz²2‘ª¨Ÿòc‡«ZŽŸôÜi‘ ª©±4ŽŸ@‘8â0ŽŽ‘ÑĵxŸûÞÿ±3´=±2Ž– ‘ʵdx‘Dz=Ÿù<$‘-~4Ž‘-~Ÿw‰feŸ (Ö5ŽŽŽŽ‘ `²µxŸûÞÿ±5´=±2Ž“Ÿñæ^«ŽŽŸóøÛ‘"ð¶±4ŽŸ,‘"ð¶0ŽŽ‘*4A²=Ÿù<$‘úK4Ž‘úKŸw‰feŸ (Ö5ŽŽŽŽ‘ f_¸‘8à²4ŸûÞÿ±5´=±2Ž‘Xâ²=Ÿù<$‘úK128Ž‘úKŸw‰feŸ (Ö‘5ŽŽŽŽ‘-µ:ŽŸÂF‘M¸ŽŽŽ‘WxŽ‘[q˲6.7‘UU#9.Ž¡‘WIf–ƒ!µf‘²(µx²)“is“an“ºo‘ÿ}'dd“²function,›Ž“that“is,˜if“µf›²(¸µx²)–k=“¸µf˜²(µx²),‘Ž“sho¸ãw‘ƒ!geometricallyަ‘Wor–UUotherwise“thatŽŸ'%’ÓCwŸòc‡«ZŽŸôÜi’ÝCx´aŽŸ@’ØÑ±·´aŽŽ’å’ʵf‘²(µx²)‘ª¨µdx–Dz=“0µ:ŽŸ}äw‘>ï5PSfile=6.7.9.eps llx=0 lly=0 urx=362 ury=362 rwi=1080ŽŽŽŸŽŒ‹?l y ý£ ? ýä‘W¹Solution:ŽŽ’ŒŽ*ºBy–“çthe“interval“addition“pr–ÿ}'op“erty–“çof“de nite“inte–ÿ}'gr“als,ŽŸ{ ’ž1|Ÿòc‡«ZŽŸôÜi’¨1}´aŽŸ@’£¿¶·´aŽŽ’°€Ïµf›²(µx²)‘ª¨µdx–Dz=“Ÿòc‡«ZŽŸôÜi‘ DZ0ŽŸ@‘UR·´aŽŽ‘kµf˜²(µx²)‘ª¨µdx–8à²+“Ÿòc‡«ZŽŸôÜi‘ 8á´aŽŸ@‘DZ0ŽŽ‘¹ùµf˜²(µx²)‘ª¨µdx:ŽŸT¦‘WºSinc›ÿ}'e–Ï µf‘²(µx²)“ºis“o˜dd,‘ÞŽµf›²(µx²)–3H=“¸µf˜²(¸µx²)º,‘ÞŽso–Ï the“ rst“inte–ÿ}'gr“al–Ï in“the“sum“is“theŽŸ&o‘Wne›ÿ}'gative–‚Yof“Ÿòc‡«ZŽŸôÜi‘ ‚Z±0ŽŸ@‘ “·´aŽŽ‘Ѭµf‘²(¸µx²)‘ª¨µdxº.‘“¼The“gr˜aph“of“µy‘"ñ²=‘ǵf‘²(¸µx²)“ºon“the“interval“²[¸µa;‘ª¨²0]ŽŸqÏ‘Wºis–cÚthe“mirr›ÿ}'or“image“in“the“µy[Ùº-axis“of“the“gr˜aph“of“µy‘"ñ²=‘ǵf‘²(µx²)“ºon“²[0µ;‘ª¨a²]º,‘mvso“theޤ ‘Wtwo‘“çinte–ÿ}'gr“alsŽŸ˜6’µ5$Ÿòc‡«ZŽŸôÜi’¿5%±0ŽŸ@’ºÃ^·´aŽŽ’Ç„wµf–²(¸µx²)›ª¨µdx;‘hŸòc‡«ZŽŸôÜi‘ i´aŽŸ@‘ª¢±0ŽŽ‘&µf“²(µx²)˜µdxŽŸ¡‘Wºc‘ÿ}'oincide–“çand“it“fol‘‚Ølows“that“Ÿòc‡«ZŽŸôÜi‘ “è´aŽŸ@‘ "!·´aŽŽ‘ã:µf‘²(µx²)‘ª¨µdx–Dz=“0º.ŽŸÆl‘M¸ŽŽŽ‘WxŽ‘[q˲6.7‘UU#11.Ž©-^‘WThe–ù’graph“of“µy‘"ñ²=–ǵxŸü^ÿ±2Ž‘|s²,› ìµx“¸“²0,˜can–ù’bšGe“considered“to“b˜e“the“graph“of“µx–Dz=“ŸùÄû¸pŽ‘ oŸùÄû‰feBøŸ;µyŽŽŽ‘_g²,Ž¡‘Wµy‘"ñ¸‘Dz0.‘qÇSho•¸ãw›UUb“y˜geometry˜that˜this˜implies˜the˜v‘ÿqÇalidit“y˜of˜the˜equationŽŸ5¦’¹;ÏŸòc‡«ZŽŸôÜi’Ã;дaŽŸ@’¾Ê ±0ŽŽ’ɼèµxŸûÞÿ±2Ž‘'µdx–8à²+“Ÿòc‡«ZŽŸôÜi‘ 8á´aŸüûr°2ŽŽŸ@‘DZ0ŽŽ‘¡ŸùH¸pŽ‘öpŸùH‰feBøŸ·ÿµyŽŽŽ‘%ädy‘"ñ²=‘ǵaŸûÞÿ±3Ž‘|sµ:ŽŸ{9Ÿ‘>ï6PSfile=6.7.11.eps llx=0 lly=0 urx=410 ury=419 rwi=1056ŽŽ¦‘W¹Solution:ŽŽ’ŒŽ*ºThe–ë rst“inte–ÿ}'gr“al›ëme“asur“es˜the˜ar“e“a˜of˜the˜p“art˜of˜the˜r“e“ctangleŽ¡‘Wunderne›ÿ}'ath–ÒLthe“curve,‘áæthe“se˜c˜ond“me˜asur˜es“the“ar˜e˜a“of“the“p˜art“ab˜ove“theŽ¡‘Wcurve.‘™–Their–“çsum“is“ther–ÿ}'efor“e–“çthe“ar–ÿ}'e“a–“çof“the“r–ÿ}'e“ctangle,–“çwhich“is“µaŸü^ÿ±3Ž‘|sº.ަ‘WCalculating–“çthe“inte–ÿ}'gr“als‘“çgivesŽŸ"8”Ÿõƒ¿’®UŸòc‡«ZŽŸôÜi’¸V´aŽŸ@’´±0ŽŽ’¿nµxŸûÞÿ±2Ž‘'µdxŽ’çš ²=Ÿù<$‘úK1Ž‘úKŸw‰feŸ (Ö3ŽŽŽŽ‘ -µxŸûÞÿ±3Ž‘|s²]ŸûÞÿ´aŽŸ™á±0ŽŽŽ’ îP²=Ÿù<$‘úKµaŸü^ÿ±3ŽŽ‘úKŸw‰fe Å£Ÿ (Ö‘bѲ3ŽŽŽŽŽŽŸ.’®UŸ÷ñÄ«RŽŸúj¦’µ;´aŸüûr°2ŽŽŸ #–’³I9±0ŽŽ’¿£8µy[ÙŸü^ÿ±1´=±2Ž‘˜KµdyŽ’çš ²=Ÿù<$‘úK2Ž‘úKŸw‰feŸ (Ö3ŽŽŽŽ‘ -µaŸûÞÿ±3´=±2Ž‘ ‘ʲ]ŸûÞÿ´aŸüûr°2ŽŽŸ™á±0ŽŽŽ’ îP²=Ÿù<$‘úK2µaŸü^ÿ±3ŽŽ‘úKŸw‰feŤŸ (Ö‘âÒ²3ŽŽŽŽ‘ó"µ:ŽŽŽŽŸ#j¦‘WºSo–“çthis“also“shows“the“sum“is“µaŸü^ÿ±3Ž‘|sº.ŽŸâÖ‘M¸ŽŽŽ‘WxŽ‘[q˲6.7‘UU#13.ަ‘WUse–UUthe“knoš¸ãwn“area“of“a“circle“to“ nd“the“v‘ÿqÇalue“of“the“in˜tegralŽŸûÒѦŸòc‡«ZŽŸôÜi’Û§´aŽŸ@’Öžà·´aŽŽ’ã_ùŸö-­«pŽ’í_úŸö-­‰fe 2Ÿ ÒSµaŸýr±2Ž‘µS¸‘8àµxŸýr±2ŽŽŽŽ’<²µdx:ŽŽŸŽŒ‹H- y ý£ ? þFg®‘>ï5PSfile=6.7.13.eps llx=0 lly=0 urx=182 ury=290 rwi=677ŽŽ©‘W¹Solution:ŽŽ’ŒŽ*ºThe–4>curve“µy‘Ei²=‘éŸ÷ª«¸pŽ‘ >çŸ÷ª«‰fe 2ŸUUµaŸýr±2Ž‘µS¸‘8àµxŸýr±2ŽŽŽŽ‘0¥5º(with“¸µa–鸓µx“¸“µaº)–4>is“the“top“halfޤ ‘W(µy‘1¸›¼X²0º)–Hof“the“cir‘ÿ}'cle“µxŸü^ÿ±2Ž‘l²+‘šùµy[ÙŸü^ÿ±2Ž‘”¤²=˜µaŸü^ÿ±2Ž‘—»º(with“r›ÿ}'adius“µaº,‘= c˜enter“²(0µ;‘ª¨²0)º).‘/ºSo“theŽ¡‘Wvalue–“çof“the“inte–ÿ}'gr“al–“çis“half“the“ar–ÿ}'e“a–“çof“the“cir‘ÿ}'cle,“soŽŸì’¾ÈkŸòc‡«ZŽŸôÜi’ÈÈl´aŽŸ@’ÄV¥·´aŽŽ’ѾŸö-­«pŽ’Û¿Ÿö-­‰fe 2Ÿ ÒSµaŸýr±2Ž‘µS¸‘8àµxŸýr±2ŽŽŽŽ’üôwµdx‘Dz=Ÿù<$‘úK1Ž‘úKŸw‰feŸ (Ö2ŽŽŽŽ‘ -µ[ÙaŸûÞÿ±2Ž‘|sµ:ŽŸã–‘M¸ŽŽŽ‘WxŽ‘[q˲6.7‘UU#16ަ‘WCompute–UUthe“indicated“deriv‘ÿqÇativ¸ãeŽŸ˜6‘^Ž9d)ŽŽŽŸù<$‘qõdŽ‘n35Ÿw‰fe 뎟 (ÖdxŽŽŽŽ‘{üžŸòc‡«ZŽŸôÜi’…üŸ±1ŽŸ@’ŠØ´xŽŽŸù<$’•—]µdtŽ’VíŸw‰feQÄŸ (Ö²1–8à+“µtŸýr±4ŽŽŽŽŽ’§Ûä².ŽŸ@‘m¹Solution:ŽŽ’¢Ž*ºSinc‘ÿ}'eŽŸ¦o’ÉjŸòc‡«ZŽŸôÜi’Ók±1ŽŸ@’Ρ¤´xŽŽŸù<$’â®)µdtŽ’Úm¹Ÿw‰feQÄŸ (Ö²1–8à+“µtŸýr±4ŽŽŽŽŽ’÷¹È²=‘Ǹ‘ª¨Ÿòc‡«ZŽŸôÜi‘ ª©´xŽŸ@‘8â±1ŽŽŸù<$‘ÑØµdtŽ‘‘hŸw‰feQÄŸ (Ö²1–8à+“µtŸýr±4ŽŽŽŽŽ‘._µ;ŽŸ@‘mºwe‘“çhaveŽŸ¦oŸù<$’Ñ©2µdŽ’ÎͤŸw‰fe 뎟 (ÖdxŽŽŽŽ’Ü— Ÿòc‡«ZŽŸôÜi’æ—±1ŽŸ@’â%G´xŽŽŸù<$’ö1̵dtŽ’íñ\Ÿw‰feQÄŸ (Ö²1–8à+“µtŸýr±4ŽŽŽŽŽ’ =k²=Ÿù<$‘ Ló¸²1Ž‘úKŸw‰felnŸ (Ö1–8à+“µxŸýr±4ŽŽŽŽŽ‘ ™ìµ:ŽŸœy‘mº(by– e›ÿ}'quation“(13)“of“¸xŽ‘‹éº6.7,‘;®or“the“\Se˜c˜ond“F‘ÿ;¼undamental“The˜or˜em“ofŽ¡‘mCalculus").ŽŸá‘_ª¬²e)ŽŽŽŸù<$‘qõdŽ‘n35Ÿw‰fe 뎟 (ÖdxŽŽŽŽ‘{üžŸòc‡«ZŽŸôÜi’…üŸ´xŸüûr°2ŽŽŸ@’ŠØ±1ŽŽŸù<$’¨â µdtŽ’‘Ê~Ÿw‰fe6ÿùŸ fŸö~“«pŽ‘ Ÿö~“‰fe,ÿøŸ mµt–8à²+“Ÿø$û¸pŽ‘ Ž7Ÿø$û‰feÕQŸÛµt“²+“1ŽŽŽŽŽŽŽŽŽ’Éýªµ:ŽŸp§‘m¹Solution:ŽŽ’¢Ž*ºNote–”±that“the“upp›ÿ}'er“limit“of“inte˜gr˜ation“is“µxŸü^ÿ±2Ž‘|sº,‘”ãnot“µxº.‘›óT‘ÿ;¼oŽ¡‘mapply–“çe‘ÿ}'quation“(13),“we“use“the“chain“rule“with“µu–Dz=“µxŸü^ÿ±2Ž‘|sº.‘™–This‘“çgivesŽŸ6Ù\Ÿè¬Ÿù<$’™$ µdŽ’–H’Ÿw‰fe 뎟 (ÖdxŽŽŽŽ’¤ûŸòc‡«ZŽŸôÜi’®ü´xŸüûr°2ŽŽŸ@’© 5±1ŽŽŸù<$’Ð÷fµdtŽ’¹ßÛŸw‰fe6ÿùŸ fŸö~“«pŽ‘ Ÿö~“‰fe,ÿøŸ mµt–8à²+“Ÿø$û¸pŽ‘ Ž7Ÿø$û‰feÕQŸÛµt“²+“1ŽŽŽŽŽŽŽŽŽŽ’ü=Ÿù<$‘× µdŽ‘úKŸw‰fe íðŸ (ÖduŽŽŽŽ‘ÆŸòc‡«ZŽŸôÜi‘Æ´uŽŸ@‘TP±1ŽŽŸù<$‘:õßµdtŽ‘#ÞTŸw‰fe6ÿùŸ fŸö~“«pŽ‘ Ÿö~“‰fe,ÿøŸ mµt–8à²+“Ÿø$û¸pŽ‘ Ž7Ÿø$û‰feÕQŸÛµt“²+“1ŽŽŽŽŽŽŽŽŽŸù<$‘^ï[µduŽ‘^ï[Ÿw‰fe íðŸ (Ö1dxŽŽŽŽŽŽ¤‚q’ü²=Ÿù<$‘S1Ž‘úKŸw‰fe;:Ÿ fŸö~“«pŽ‘ Ÿö~“‰fe1:Ÿ mµu–8à²+“Ÿø$û¸pŽ‘ Ž7Ÿø$û‰feò]ŸÛµu“²+“1ŽŽŽŽŽŽŽŽŽ‘B o¸‘8à²2µxŽŽ¡’ü²=Ÿù<$‘ µÖ2µxŽ‘úKŸw‰feD.3Ÿ û¶ŸöC«pŽ‘ ŸöC‰fe:.2Ÿ û½µxŸýr±2Ž–µS²+‘8àŸ÷ª«¸pŽ‘ Ž7Ÿ÷ª«‰felnŸUUµxŸýr±2Ž“²+‘8à1ŽŽŽŽŽŽŽŽŽ‘I[±µ:ŽŽŽŽŽŸŽŒ‹ S1 y ý£ ? ýä‘M¸ŽŽŽ‘WxŽ‘[q˲10.9‘UU#2.Ž©i¹‘WClearly‘ÿ*ª,ŽŸì’ÓÆ3Ÿòc‡«ZŽŸôÜi’ÝÆ4´ŽŸ@’ÙTm±0ŽŽ’äÞK²sinŽ’òеx‘ª¨dx–Dz=“2µ:ŽŸKj‘W²Calculate–ithe“v‘ÿqÇalue“of“this“inš¸ãtegral“appro˜ximately“using“Simpson's“ruleޤ ‘Wwith‘UUµn–Dz=“4.ŽŸœ‘W¹Solution:ŽŽ’ŒŽ*ºDivding– the“interval“²[0µ;‘ª¨[Ù²]“ºinto“µn–Å ²=“4– ºe‘ÿ}'qual“subintervals,‘C"theŽ¡‘Wendp–ÿ}'oints‘“çar“eޤØnŸùY‘uê´µxŸÿ±0Ž‘C‹²=‘Ç0µ;Ž’Ÿ:¯xŸÿ±1Ž‘C‹²=‘ǵ[Ù=²4µ;Ž’ÚxŸÿ±2Ž‘C‹²=‘ǵ[Ù=²2µ;Ž’o xŸÿ±3Ž‘C‹²=‘Ç3µ[Ù=²4µ;Ž’IDwxŸÿ±4Ž‘C‹²=‘ǵ[Ù²;‘>ºsoŽŽŽŸ ¬‘uê´µyŸÿ±0Ž‘C‹²=‘Ç0µ;Ž’Ÿ:¯yŸÿ±1Ž‘C‹²=‘ÇŸ÷ºP¸pŽ‘ oŸ÷ºP‰feŸE°²2ŽŽŽ‘pµ=²2µ;Ž’ÚyŸÿ±2Ž‘C‹²=‘Ç1µ;Ž’o yŸÿ±3Ž‘C‹²=‘ÇŸ÷ºP¸pŽ‘ oŸ÷ºP‰feŸE°²2ŽŽŽ‘pµ=²2µ;Ž’IDwyŸÿ±4Ž‘C‹²=‘Ç0µ:ŽŽŽŽ¡‘WºSimpson‘þúP's–“çrule“givesŽŸ)ˆ‘WŸòc‡«ZŽŸôÜi‘a´ŽŸ@‘\Ž<±0ŽŽ‘h²sinŽ‘v àµx‘ª¨dx‘ǸŸù<$‘ò½µŽ‘úKŸw‰fe Ÿ (Ö²12ŽŽŽŽ‘-€(0–·x+“4“¸“Ÿ÷=VpŽ‘ ÏŸ÷=V‰feŸª²2ŽŽŽ› е=²2“+“2“¸“²1“+“4“¸“Ÿ÷=VpŽ‘ ÏŸ÷=V‰feŸª²2ŽŽŽ˜µ=²2“+“0)–Ç=“(1–·x+“2Ÿ÷=V¸pŽ‘UWŸ÷=V‰feŸª²2ŽŽŽ‘ UX)Ÿù<$‘33µŽ‘33Ÿw‰feŸ (Ö‘‡²6ŽŽŽŽ‘ <œ¸‘Dz2µ:²00456µ:ޤˆ•‘M¸ŽŽŽ‘WxŽ‘[q˲10.9‘UU#5.ަ‘WThe–UUexact“v‘ÿqÇalue“ofŽŸ¦o’ߊ©Ÿòc‡«ZŽŸôÜi’銪±2ŽŸ@’åã0ŽŽ’ï±ÅµeŸûÞÿ·´xŸüûr°2ŽŽ‘0µdxŽŸKj‘W²is–y@not“kno¸ãwn,‘‚;but“to“10“decimal“places“it“is“0.8820813908.‘݈Calculate“thisŽŸ ‘Win•¸ãtegral›UUappro“ximately˜using˜Simpson's˜rule˜with˜µn–Dz=“4.ŽŸœ‘W¹Solution:ŽŽ’ŒŽ*ºWe‘“çhaveŽŸÍëŸù©‘xn³µxŸÿ±0Ž‘C‹²=‘Ç0µ;Ž’¡¾®xŸÿ±1Ž‘C‹²=‘Ç1µ=²2µ;ޒ܏wxŸÿ±2Ž‘C‹²=‘Ç1µ;Ž’œéxŸÿ±3Ž‘C‹²=‘Ç3µ=²2µ;Ž’GÏ•xŸÿ±4Ž‘C‹²=‘Ç2;‘>ºsoŽŽŽŸ z‘xn³µyŸÿ±0Ž‘C‹²=‘Ç1µ;Ž’¡¾®yŸÿ±1Ž‘C‹²=‘ǵeŸü^ÿ·±1´=±4Ž‘Ñ˵;ޒ܏wyŸÿ±2Ž‘C‹²=‘ǵeŸü^ÿ·±1Ž‘ ¼tµ;Ž’œéyŸÿ±3Ž‘C‹²=‘ǵeŸü^ÿ·±9´=±4ŽŽ’GÏ•µyŸÿ±4Ž‘C‹²=‘ǵeŸü^ÿ·±4Ž‘ ¼tµ:ŽŽŽŽŸ¿±‘WºSimpson‘þúP's–“çrule“givesŽŸ¨Ò‘WŸòc‡«ZŽŸôÜi‘a±2ŽŸ@‘\Ž<0ŽŽ‘g'µeŸûÞÿ·´xŸüûr°2ŽŽ‘0µdx›Ç¸Ÿù<$‘úK²1Ž‘úKŸw‰feŸ (Ö6ŽŽŽŽ‘ -(1–&æ+“4“¸“²0µ:²7788“+“2“¸“²0µ:²3679“+“4“¸“²0µ:²1054“+“0µ:²0183)˜¸˜²0µ:²8818µ:Ž¡‘M¸ŽŽŽ‘WxŽ‘[q˲10.9‘UU#9ަ‘WThe–ÿ width,›Nin“feet,˜at“equally“spaced“pGoinš¸ãts“along“the“fairw˜a˜y“of“a“hole“onޤ ‘Wa–þùgolf“course“is“givš¸ãen“in“Fig.“10.7.‘n³The“managemen˜t“wishes“to“estimateŽ¡‘Wthe›èn•¸ãum“bGer˜of˜square˜y“ards˜of˜the˜fairw“a“y˜as˜a˜basis˜for˜deciding˜ho“w˜longŽ¡‘Wit–{íshould“takš¸ãe“a“groundsk˜eepGer“to“mo˜w“it.‘åŽUse“Simpson's“rule“to“pro˜videŽ¡‘Wsuc¸ãh–UUan“estimate.ŽŸœ‘W¹Solution:ŽŽ’ŒŽ*ºApplying–Simpson‘þúP's“rule“with“µn–©Ö²=“12–ºgives“the“fol‘‚Ølowing“ap-Ž¡‘Wpr›ÿ}'oximation–“çof“the“ar˜e˜a:ŽŸ8Ÿù<$‘X35²1200Ž‘X35Ÿw‰feŸ (Ö‘36ŽŽŽŽ‘mfl(0+4¸²180+2¸²190+4¸²175+2¸²170+4¸²185+2¸²195+4¸²205+2¸²210+4¸²200+2¸²195+4¸²170+0)ŽŸÞl‘W¸–“çº212,667“squar›ÿ}'e“fe˜et,“or“23,630“squar˜e“yar˜ds.ŽŽŸŽŒø^¨ƒ’À;èy— óý': cmti10óò"V cmbx10ó !",š cmsy10ó O!â…cmsy7ó  b> cmmi10ó 0e—rcmmi7óKñ`y cmr10óÙ“ Rcmr7ó†›Zcmr5óú±u cmex10ùjãßßßßß