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by: Reyes Glover


Reyes Glover
GPA 3.67

Louiza Fouli

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Louiza Fouli
Class Notes
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This 11 page Class Notes was uploaded by Reyes Glover on Sunday September 6, 2015. The Class Notes belongs to M 408L at University of Texas at Austin taught by Louiza Fouli in Fall. Since its upload, it has received 17 views. For similar materials see /class/181424/m-408l-university-of-texas-at-austin in Mathematics (M) at University of Texas at Austin.

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Date Created: 09/06/15
M 408L Integral Calculus Prepared by Chris Mirabito Test 2 Review Sheet Cookbook March 20 2007 Test 2 will be held on Tuesday March 27 2007 at 700pm in GSB 2124 THIS IS A DIFFERENT LOCATION THAN EXAM 1 The test will over 81 82 83 84 85 87 88 153 161 162 and 163 in the course textbook1 You cannot use a calculator or any other electronic devices books or notes in any form The test will be 2 hours long and will be worth 20 of your nal course grade On the test will be up to 20 multiplechoice questions similar to what you have found on Homeworks 5 through 9 Unlike the homework problems you will NOT lose points for guessing Please bring your UT ID cards to the test We will be checking them your cooperation with the University policy on cheating is expected Also rem ember to bring a 2 pencil and eraser to the test as none will be provided All other testing materials will be provided This review sheet is intended to help you prepare for the test It states in a nutshell pretty much everything that we have covered in the course since Test 1 but this list is not exhaustive Please be sure you are familiar with all concepts listed here If you are having trouble please stop by Dr Fouli s or your TA s of ce hours Do not wait until the last minute to ask for help And remember material needed for Test 1 could very well be used again on this test that means volumes exponentials etc Darice Chang will hold a review session Friday March 23rd 2007 from 200pm until 400pm in BUR 112 1 highly recommend attending this since it will help answer any lastminute questions you may have There is a practice test for you attached at the end It is in a format similar to the actual test You should take this test in a quiet area without books notes and calculators Hopefully it will help you study 31 Students with legitimate documented learning disabilities or special needs should consult the First Day Handout located at http www ma utexas edudevmathCoursesM4OSLhandout html for information regarding testing dates times and locations which will be different from that of the regular time Students who cannot take the test at the regularly scheduled time due to con icts with other classes exams or docu mented illnesses should also consult this page for makeup test information In all cases make sure Dr Fouli knows about your situation well in advance 81 Integration by Parts 0 Memorize this formula udv uvi vdu b b 7 vdu a a o It helps to make a little table for u du v and dv After lling in this little table all you have to do is plug n chug o How to choose u LIATE b o For de nite integrals udv W a L ogarithmic functions I nverse Trigonometric functions 1If you have another version or a previous edition ofthe textbook the sections to you need to know may be slightly different If you are not sure which sections you should study please ask me A lgebraic ie polynomials radicals etc T rigonometric functions E xponentials 0 Your choice of u and dv should make the resulting integral on the right hand side easier to solve not harder If you nd that your resulting integral is harder to solve than the original you probably made the wrong choice of u and dv Try a different choice instead o It is a good idea to select it such that it s easy to take its derivative and dv such that it is easy to integrate 82 Trigonometric Integrals o This is a tricky section that will require some cleverness on your part Study this section very carefully and proceed with caution You need to know trigonometric identities in order to do well in this section 0 Study the boxes on pages 520 and 522 which describe some basic strategy for integrating sinmxcosnxdx and tanmxsec xdx o In general you shouldn t try to use usubstitution until you simplify the integrand using 1 or more trigonometric identities o If you have sinmxcos xdx l lfm is odd split off a copy of sinx let sinzx l 7 cos2 x then let u cosx 2 lfn is odd split off a copy of cosx let cos2x l 7 sin2 x then let u sinx 3 If both powers are odd pick your poison l 7 cos2x l cos 2x and cos2x 2 4 If both powers are even let sinzx o lfyou have tanmxsec xdx l lfm is odd split offa copy of secxtanx let tan2x sec2x7 1 then let u secx 2 lfn is even let sec2x l tan2 x then let u tanx 3 lfm is odd and n is even pick your poison 4 lfm is even and n is odd try using the identity taan sec2x 7 l 0 Remember that tanxdx lnl secxl C and secxdx lnl secxtanxl C o If you get stuck and nothing else seems to be working don t be afraid to multiply the integrand by factors like s secx tan 7 You Will need to do this when integrating secx or cscx for example secx secx tanx 83 Trigonometric Substitution 0 You need to know trigonometric identities to do well in this section too 0 The basic idea behind this If you see Substitute and simplify using 1127x2 xasin9 l7sin29cos29 xainrx2 xatan9 ltan29 sec29 xx27ai2 xasec9 sec297l tan29 0 Remember that you can use this when you see quantities like x2 a232 or x2 a223 too 0 Remember also to draw a triangle to gure out the values for the other trigonometric functions after you have integrated 0 Expression not in one of the forms above Try completing the square and factoring 84 Integration of Rational Functions by Partial Fractions 0 Before trying to use partial fractions try to simplify if degnumerator 2 degdenominator you must divide using polynomial long division 0 Factor all terms in the denominator if possible 0 The 4 cases to consider and what to do A B l Distinct LinearFactors Split into form m m 2 Re eated Linear Factors S lit into form A B i p I p ax b ax b 2 A B C D 3 Distinct Quadratic Factors Split into form zx m Ax B Cx C 4 Repeated Quadratic Factors Split into form m m 0 NOTE In cases 3 and 4 the quadratic factors must be irreducible This will not work if the factors are reducible o Often you ll have mixtures of these cases Just remember to treat each factor separately For example 1 7 A B Cx D Ex F 4x525x2 6x72 T 4x5 4x 52 5x2 6x7 5x2 6x72 85 Strategy for Integration 0 It is important to read the book andor your notes here It gives details on what to look for in integrands o The basic strategy is i Simplify the integrand if possible 19 Try using usubstitution L Look at the integrand Do you have trigonometric functions like in 82 rational functions like in 84 or a product to 2 things with 1 easy to differentiate and the other easy to integrate do integration by parts here or things like x2 i a2 do trigonometric substitution here gt Try again ie you ll have to be a little more crafty 87 Approximate Integration 0 We have 5 methods of approximating an integral the Left Endpoint Rule the Right Endpoint Rule the Mid point Rule the Trapezoid Rule and Simpson s Rule Some are more accurate than others do you remember which ones I said were best 0 Trapezoid Rule fx0 2fx1 2fxn71 fltxnl Ax o Simpson s Rule fx0 4fx1 2fx2 2fxn2 4fxn1 fxn Remember that n must be even here Make sure you know the order of the coef cients Draw the mountains to help you 0 You do NOT need to know the formulas for the error estimates for the Trapezoid Rule and Simpson s Rule 0 Many people asked me whether they would need to evaluate the approximation to a decimal No need to freak out here The choices will be expressions for the approximation not decimals unless they re really easy to evaluate 88 Improper Integrals o How to set them up m I 1 fx dx lim fx dx 5 b 2 fx dx my fx dx m p z 3 fx dx lim fx dx lim fx dx M saw 5 aw P o In the setup above pick 7 to be something convenient like 0 o If f x has a vertical asymptote check for this split the integral there and then take left and right hand limits 0 If you split an integral and one portion diverges stop The integral diverges 0 Remember to carry the limits all the way through Not doing this will confuse you 153 Partial Derivatives 0 Do not let notation confuse you ifz fxy then 3 3 3 my Emmy g f1D1fDxf and 8 8 8 13x7yfv 7 37mm fz szDyfA To nd differentiate with respect to x and treaty as a constant And to nd differentiate with respect to y and hold x constant 0 What if you have uxyz7 wv7 9 a y 5 S and you wanted ax Differentiate with respect to x and hold all other variables constant 0 Remember the physical meaning of a partial derivative ab is the slope of f in the xdirection at ab and similarly for y o Clairaut s Theorem y fy as long as f and all its partial derivatives are continuous In other words if f fy y and fy are continuous then you can switch the order in which you differentiate and you ll get the same answer 161 Double Integrals over Rectangles 0 Remember that the Fundamental Theorem of Calculus both parts holds for functions with 2 or more variables too All the properties we learned in 5l and 52 still work 0 You can also use the same methods for approximating a double integral as before Left Endpoints Right End points etc This time though it s possible to use Left Endpoints along x and Right Endpoints along y etc 162 Iterated Integrals b d o The expression f x7 y six dy is an iterated integral To solve these integrate with respect to one vari a c able at a time inside out b d d b o Fubini s Theorem If fxy is continuous then fxydxdy fxy dydx In other words as long as f is continuous you can change the order 61f iiftegration But rem embzer switching six and dy means you must change the limits of integration too 0 As a reminder as before you can pull constants out of both integrals But if you re integrating with respect to say x rst and your integrand is something like yf you can only pull the y out of one of the integrals not both 163 Double Integrals over General Regions 0 When you change the order of integration you must change your limits so that the ones on the inner integral are in terms of the other variable 0 It is helpful to draw the region that you are integrating over If you choose to integrate with respect to x rst draw a horizontal line anywhere through the region If you choose to integrate with respect to y rst draw a vertical line anywhere through the region 0 Your limits on the inner integral will be functions The lower limit is where you enter the region and the upper limit is where you leave the region 0 The limits on the outer integral will always be numbers You should never have variables in your limits of integration on the outerintegral Ever Ever GOOD LUCK If you have any lastminute questions em ail me or Dr Fouli M 408L Integral Calculus Test 2 Practice Problems March 20 2007 This practice test is intended to help you study for Test 2 It includes 17 multiplechoice questions which may be similar to What you may be asked to do on the actual test You should complete these questions on your own in a quiet area Without books notes or any electronic devices You should give yourself 2 hours to do this practice test so you can learn to pace yourself about 6 minutes per problem Hopefully these questions will be harder than the ones that will be on the test Use a separate sheet of paper for your work A page to mark your answers and a solution page are located on the last page 1 tan5xdx a xtan4 x 7 5xtan4x sec2 x C tan6 x 6 C 0 l l d Etan4x7 Etan2xlnlsecxl C C 3 e tar x 7tanxlnl secxl C 2 sin4xcos4xdx sin2x C 16Jr a x sin2x b T C sin8x C 2 C sin4 2x sin4 4x d 7 7 C 16 64 3 sin4x sin 8x 6 128 128 1024 C 3 2coslnxdx a 2xsinlnx cos lnx C b 2xsinlnx 7 cos lnx C c xsinlnx 7 cos lnx C d xsinlnx cos lnx C e xcos lnx 7 sinlnx C 12 4 8sin 1xdx 0 a 271 42 7 b 2n 7 42 c n7 42 7 2H d g 42 2n e g 7 42 7 5 3 sinzxcos3 xdx a Divergent 3 b s1n3x7 3 s1n5xC 3 0 Sin3 x 3 Sin5 xC 3 d g cos3x7 s1n5xC 3 e 73 s1n3x7 cos5xC 6 Which of the following integrals on the interval 0 has the greatest value 7r4 a sintdt 0 7r4 b cosldl 0 7r4 c cos2 Idl 0 7r4 d cos 2 dt 0 7r4 e sinl cost dl 0 l 3 m a 3tan71ltx72gtC b 3sin 1 2 C c 3tan71ltx23gtC d sin 1ltx2gtC e tan71ltxJ2r3gtC Find a solution to the rstorder differential equation Q7 2 dx 7 x7lx32 so with an unrestricted initial condition O a yx 7 lnlxilllnlx3li C b yx 7 lnlxilli lnlx4r3l4r 4rC c yx 7 lnlxilllnlx3l C d yltxgt m1x7117 m1x31273c e yltxgt m1x7117 m1x31873c The Dodge Viper SRT10 can go from 0 to 60 mph in less than 4 seconds Suppose velocity was measured according to the following table 18 v0 mph 0 0 05 0125 1 1 15 3375 2 8 25 15625 3 27 35 42875 4 64 Use Simpson s Rule to determine the total distance traveled by the Viper ie the lengths it will go to try to get you to spend 8574500 on a car 1 m l m l m l m A 37 40125 21 43375 28 415625 227 442875 64 miles 40125 21 43375 28 415625 227 442875 64 miles 20125 41 23375 48 215625 427 242875 64 miles 20125 21 23375 28 215625 227 242875 64 miles 721W0125 1 3375 8 15625 2742875 64 miles 3 L o 1 7102 a 7 b 7 c 1 d 2 3 2 e D ivergent 11 L 4 Lquot lfz xsecy then Which of the following integrals on 01 X 01 1ln4x 7d A we x a Divergent b ln472 c 2ln472 d 4ln41 e 4ln471 A key measure of a nation s economic strength is the rate of in ation In 2005 the US in ation rate was measured at 4 per year meaning 104 today could buy 1 worth of goods in 2005 Now typically the in ation rate is measured monthly but suppose that it is possible to measure the in ation rate continuously Suppose also that in the longterm the Fed manages to stabilize the in ation rate so that the in ation rate 11 0046 cos I where I is measured in years and I 0 corresponds to 2005 Assuming thatII continues on this trend forever which best describes the longterm US economic trend a 102 will buy 1 worth of 2005 goods 11 098 will buy 1 worth of 2005 goods c 1 will buy 1 worth of 2005 goods d Meltdown for the US monetary system e Cannot tell from the given data 1ffxyz sinxycosyz then fyzxy0 a it 77 b 5 C 1 d 1 e 0 a BxBy a 0 b cscy c seczy d secytany e tanyil has the smallest value 1 1 a lnxlnydxdy 0 0 1 1 b my dxdy 0 0 c 0101xydxdy d A1 1 WWW l CW1 y 3 2 3w2 16 7d d A 1216x2 y x 18813 mm VV N 17 KB is the triangle with vertices at 02 11 and 32 then y3 dA D 3 a E b 1 2 3 4 5 6 7 8 9 10 H 12 13 M 15 16 17 ANSWER SHEET


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