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WUSTL MATH 132 - m132_E2cF10

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MATH 132 EXAM II FALL 2010 This exam should have 18 questions. Part I will have 16 multiple choice questions, 5 points each. Part II will have 2 hand graded questions, 10 points each. Please check to . If you do not have a to mark your card,to see that your exam is complete pencil please ask to borow one from your proctor. Write your (not your SS number) on the six blank lines on the top ofID NUMBER your answer card, using one blank for each digit. After being sure of which answer you think is right, shade in the corresponding box on your card.1) Evaluate '-9= ÐBÑ .BÞ$EÑ -9= ÐBÑ ! -9= ÐBÑ ! G#%"%FÑ -9=ÐBÑ ! -9= ÐBÑ ! G"$$GÑ B " -9=ÐBÑ ! G""##HÑ =38 ÐBÑ " =38 ÐBÑ ! G#%"%IÑ =38ÐBÑ " =38 ÐBÑ ! G"$$JÑ B ! =38ÐBÑ ! G""##KÑ -9= ÐBÑ ! G"%%LÑ -9=ÐBÑ " -9=ÐBÑ =38 ÐBÑ ! G#MÑ =38ÐBÑ " =38ÐBÑ -9= ÐBÑ ! G#NÑ =38ÐBÑ" -9= ÐBÑ ! G""#$#2) What is the form of the integral after we make the trigonometric'B""B##È.Bß substitution, Bœ=38Ð ÑÞ)EÑ =38Ð Ñ .'))FÑ -9=Ð Ñ .'))GÑ =38 Ð Ñ .'#))HÑ - 9= Ð Ñ .'#))IÑ =38Ð Ñ-9=Ð Ñ.')))JÑ =38 Ð Ñ-9=Ð Ñ.'#)))KÑ =38Ð Ñ -9= Ð Ñ .')))#LÑ >+8Ð Ñ .'))MÑ =/- Ð Ñ .'#))NÑ =/-Ð Ñ>+8Ð Ñ.')))"$#0"!Þ Þ#2$Ñ .B Ð B # !ÑÞUse to evaluate partial fractions'B ! #B ! B#EÑ 68Ð#B ! "Ñ ! GFÑ 68ÐB ! BÑ ! G#GÑ +<->+8ÐBÑ ! GHÑ ! G"+<->+8ÐBÑIÑÐ#B ! "Ñ#JÑ68Ð Ñ ! GB!"BKÑ 68Ð Ñ ! GBB!"#LÑ 68 Ð Ñ ! G#BB""MÑ 68Ð+<->+8ÐBÑÑ ! GN Ñ +<->+8Ð6+8ÐBÑÑ ! G4) Approximate using the Trapezoidal Rule with to 4 decimal places'"#"B.Bß 8 œ %ß ÞEÑ !Þ'* ! FÑ !Þ'*'" GÑ !Þ'*&$ HÑ !Þ'*%! IÑ !Þ'*$# J Ñ !Þ'*##5KÑ !Þ'*"( LÑ !Þ'*"" MÑ !Þ'*(! N Ñ !Þ'*!!"$#0"!Þ#Þ$5) How large should n be to guarantee that Simpson's Rule of approximation for is acurate to within 0.0001 ? ( recall |E'!#"#BOÐ,"+Ñ")! 8/.B À lŸ ÑW&%EÑ( FÑ) GÑ* HÑ"! IÑ"" JÑ"# KÑ"$ LÑ"% MÑ"& NÑ"'6) Evaluate the improper integral if it is convergent. If not say it is'!#"B"#"B/.Bß divergent.EÑ FÑ GÑ / HÑ / IÑ J Ñ / KÑ " "LÑ " """ " " "/ ///#$Î#/È È#$Î#ÈMÑ / " "NÑ.3@/<1/8>#"$#0"!Þ#Þ%7) Find the area of the region enclosed by the curves y = x##+8. C œ " B ! %B ÞA) "$B) #$C) "#D) $#E) %$F) &#G) &$H) )$I) "!$NÑ"$$8)Find the area of the region in the first quadrant enclosed by and by the lines Bœ# Cß Bœ$" Cß B œ !ÞÈA) "$B) #$C) "#D) $#E) %$F) &#G) &$H) )$I) "!$NÑ"$$"$#0"!Þ#Þ59)Find the volume of the solid obtained by rotating the region bounded by the curve y = 2 and the line y =x .ÈB about the x-axisA) 1B) "1 51GÑ"# 1HÑ20 31IÑ#%1JÑ$2 31KÑ $# 1LÑ %! 1MÑ%3 51NÑ%) 110)Find the volume of the solid obtained by rotating the region bounded by the curve and the line .BœC Bœ" Bœ"#about the lineA) 1B) ""1 51GÑ# 1HÑ2 "&1IÑ%1JÑ$"&2 1KÑ$& 1LÑ %! 1MÑ"'" 51NÑ%) 1"$#0"!Þ#Þ6""Ñ Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the curve and the lineCœ$! #B " B# B ! Cœ$ Þabout the -axisCA) B) G11 1#( ) $ $!#$&( 2 2 1111GÑ# HÑ IÑ% JÑ ÑLÑ %! MÑ N Ñ %) 11$&% 1"$#0"!Þ#Þ712) Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the curve and the lines Cœ B Cœ!ß Bœ"ßÈ .about the line Cœ " "A) B) 11 1""$$'& "&&1 2 2 11 11GÑ# HÑ IÑ % JÑ KÑLÑ %! MÑ N Ñ %) 11"$) 1"$#0"!Þ#Þ)13) Which of the following integrals gives the length of the curve ?Bœ>" >ßCœ> ß!Ÿ>Ÿ$# $Î#%$A) 'È!$#$ ! %> .>FÑ " ! %> .>'È!$#GÑ $ ! &> .>'È!$#HÑ " ! &> .>'È!$#IÑ $ ! $> .>'È!$#JÑ "! $> .>'È!$#KÑ $ ! >.>'È!$#LÑ " ! >.>'È!$#MÑ $ ! #> .>'È!$#NÑ "! #> .>'È!$#14) Find the average value of y = x over the interval [ 1 , 3 ] .#A)"!$FÑ""$GÑ%HÑ"$$IÑ"%$JÑ&KÑ"'$LÑ"($MÑ 'NÑ"*$"$#0"!Þ#Þ*15) A force of 10 lb is required to hold a spring stretched 5 inches beyond its natural length . Find the amount of work done , in ft-lbs , to stretch it from its natural length to 6 inches beyond its natural length .A) 1B) 2C) 3D) 4E) 5F) 6G) 7H) 8I) 9J) 1016) Suppose 3 ft-lbs of work is required to stretch a spring from 2 ft to 4 ft beyond its natural length . How far , in ft , will a weight of 5 lb stretch the spring , beyond its natural length ?A) 20B) 18.75C) 15D) 10E) 8F) 6.25G) 5H) 3.5I) 2J) 1.5"$#0"!Þ#Þ"! PART II: CLEARLY WRITE YOUR SOLUTION AND HOW YOU GOT IT. Name : _______________________ ID # : ______________ Subsec:_______17) Consider the region in the bounded by the curve andfirst quadrant CœBß# the lines and .#B ! Cœ) Cœ!a) Find the area by . ( 5 points)integrating with respect to x! ‘'+,Ð0ÐBÑ "1ÐBÑÑ .B ( You more than one integral.)might needb) Find the area by . y y y ( 5 points)integrating with respect to y! ‘'cdÐ0Ð Ñ "1ÐÑÑ. (Make sure you get the same answer, as above.)"$#0"!Þ#Þ"" PART II: CLEARLY WRITE YOUR SOLUTION AND HOW YOU GOT IT. Name : _______________________ ID # : ______________ Subsec:_______1 ) Consider the solid obtained by the region bounded by) rotating about the x-axis y = x#+8. C œ #B Þa) Find the volume of the solid ( 5 points) .using the disc/washer method b) Now use the to find the volume of the abovemethod of cylindrical shells solid and see that you get the same answer ( 5 points )


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WUSTL MATH 132 - m132_E2cF10

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