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Calculus II

by: Braeden Lind

Calculus II MA 241

Braeden Lind
GPA 3.93

R. Kalhorn

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R. Kalhorn
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This 57 page Class Notes was uploaded by Braeden Lind on Thursday October 15, 2015. The Class Notes belongs to MA 241 at North Carolina State University taught by R. Kalhorn in Fall. Since its upload, it has received 9 views. For similar materials see /class/223706/ma-241-north-carolina-state-university in Mathematics (M) at North Carolina State University.

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Date Created: 10/15/15
Sectlon b 1 Instructor Rebecca Kalhorn Mole About Aleas Consider We want to nd the area between two curves where one curve the graph ol mt rs above the other the graph ol m on the mtelwl do mple Consrder mt 4e and m A Exzt We would lrke to nd the area ol the regron on the plot below Notroe that m 2 We lor all 0 e 72 0 We know how to nd the area below we and the area below We usrng rntegrals but how do we nd the area ol the regron between the two curves Oloourse we can usrn ol equal and dlf felent As the number ol our approxlrnatlon gets better In act as the number ol our approxrrnatron Recall The area below a contlnuous lunctlon f between the llnes a and at 17 ls de ned to be be rs a selected pornt taken lrorn the a subrnterval ol u b Section 6 1 Instmctox Rebecca Kalhoxn Area Between Curves Let s be the region bounded by the graphs for all a0 6 W The area of the xeglo of continuous functions f and y where at 2 m n s is A Example Fmd the area enclosed by the graphs of 3 me 7 a0 and 3 3e 7 2 A Example Fmd the area of the region enclosed by the gaphs of y2 a0 and a0 7 3 a Sectxoh 6 1 Instmctox Rehem Kalhom A Example Fmd the area of the region enclosed by the graphs of M 2 and 2 W0 1 A Example Fmd the area of the region enclosed by the graphs of m 4 a 00 Secllon b 2 Inslxuclox Rebecca Kalhoxn Volumes C nsider We wanl lo nd llne volume ol a solld llnal ls nol a common slnape In some cases we can use Calculus lo do llus De nition A wlnloln m llus class we wlll Just refer lo as a s a solld bounded by a plane region 3 called llne and a conguent leglon 32 looaled ln parallel plane The oonslsls ol all polnls on llne segmenls llnal ale perpendlcular lo llne base and Joln 3 lo 32 Volume ol wnal u llne sold s s nol a cyllndel A Fusl we wlll dwlde oul volume lnlo n quotslabsquot ol equal wldlln We lnen approxlmate llne volume ol eaoln slab uslng a By up llne ap ploxlmated volumes ol llne slabs we gel an approxlmatlon ol llne lolal volume The approxlmatlon gels bellel as llne number ol slabs In fut llne vol ume ls lhe ol lhe sum as n Sectroh 6 2 Instructor Rehem Kalhorh a0 7w1 s Example 1 Cohsrder the area bounded by the graph of at and a0 1 Rotate thrs graph about the wax re j u u u n m M 39 m We wru approxrrhate the volume by shorhg the sohd rhto quotslabsquot ofequal herght and estr Bks whroh are We 39 o the 39 oyhhders so A ofeach A 39 Smce the rs where r rs the For any gweh ohsk what rs the raohus What rs M So we have V Q But thrs rs Just a so as n approaches rh ruty we get the Sectloh b 2 Instructor Rebecca Kalhorh Example 2 Consldel the area bounded by the graph ofg 21 or 1 and a0 1 Rotate thre graph about the reams and nd the volume ol the solld ol revolutroh A Example 3 Consldel the area bounded by the graph of o a o 1 and o 2 Rotate thre graph about the yraxls and nd the volume ol the solld ol revolutroh Sectloh 6 2 Instructor Rehem Kalhom Example 4 Recall the example lrom Seotroh s 1 Consldel M 4e and m For all a0 6 72 0 at 2 m We would llke to nd the area ol the plot below resultmg solrd The resultmg solld would look llke the lollowmg Notlce that the solld ls more llke a ll we use asks to nd the V017 ume ol the solld our dlsks wlll have We call these risks wlth What ls the volume ol a washer Draw a represehtatrve reotahgle oh the graph above What s the volume rotatrhg thrs rectangle Seotxoh 6 2 Instxuotox Reheooa Kalhom So what is the volume of our sohd A Example 5 Consider the area bounded by the graph of a0 y e 2 2y 7 a0 7 2 0 a a 1 Rotate this graph about the wraxis and nd the volume of the sohd of revolution A may 3 Sectlon b 2 Instructor Rebecca Kalhorn s Example 7 Consrder the area bounded by the graph ol o 2e 7 2 and the wraxls Rotate thrs graph about the yraxls and nd the volume ol the sohd ol revolutron Note Thrs example rs hard to do usrng washer method Why We need to know the volume ol a llke the lollowlng We can see the volume ol the Draw a representatrve reotangle ln the gaph above Now we wlll rotate that rectangle to get a shell What rs the radrus7 What rs the herght7 What ls the wldth7 So the volume ol the solld wlll be Section 6 2 Instructor Rebecca Kalhom Helpiul webslte ior Method oi cvllnolrloal Shells http matholemos gcsu eolumatholemosshellmethool Question How do I know which method to select ollslr washer or shell7 Answer There ls no answer to thls questlon The only way to know which ls the best and L ll la th and then using the Alter that decide which method gves the least number oi andor which method gives the lntegranol that ls to lntegrate Below ls a summary that mlght he p Summary Calculatlng volumes oi revolutlon when the axls oi revolutlon ls elther the wraxls or the yraxls s Disk Method A representatlve rectangle and thus ollslr ls to the axis oi revolutlon The axis oi revolutlon ls a oi the xegon Horizontal Axis of Revolution Veztical Axis of Revolution 5 e 3 3 2 2 0206 12162 unzuallzlaz helght oi ollsk ls helght oi ollsk ls v V Section 6 2 Instructor Rebecca Kalhom s Washer Method A representative rectangle and thus washer is Horizontal Axis of Revolution s Shell Method A representative rectangle and thus cylmdncal shell is to the axis oi revolution Horizontal Axis of Revolution Vertical Axis of Revolution 4 3 3 z 2 1 l nzua HIKE gm 02 us 1173916 2 Width ol cylmdncal shell is with ol cylmdncal shell is Section 62 Instructor Rebecca Kalhorn Example 8 Use Cylindrical Shell Method to nd the volume of rotating the area bounded by x axis7 z 17 z 27 and the graph of y x3 Final Note Disks Representative rectangle is perpendicular to axis of revolution Shells Representative rectangle is parallel to axis of revolution Seotlon 6 3 Instructor Reheooa Kalhorn Ala Length C nsider We want to nd the length ol a curve usrng Calo ulus By length we mean the measurement we would get u we put a strrng on the curve strarghted the strrng out and measured rt Applications Smce the length ol a curve can be used to represent drstanoe traveled work etc ndrng the length ol an arc can be very uselul Consrder the lollowrng curve I We can approxrmate the length ol the curve by splrttrng up the rnterval o to b rnto equal subrntervals ol length M and approxrmatrng the length ol the curve on those subrntemls wlth stralght llnes y dPa1rPt represents the drstanoe between the pornts PH and Pu and n denotes the numr her ol subrntervals then we have L Seouon 6 3 Insuuolox Rebeooa Kalnoxn Let s examlne d024 R Thls IS the dlsLanoe between two polnls along a stxalght llne dual a Nouoe Ayg Recall WW m As Am A 0 and thus as n A oo th5 becomes equallw Now we have MPH a m Thexeloxe L The analogous argument works for a lunonon m from 3 o to y What n the curve ls parametnc Conslder the lollowlng curve wht A7t mgtgs Section 63 Instructor Rebecca Kalhorn Find the points P0 through Pn by splitting up the interval 735 into equal subintervals with endpoints t07t17 7tn Therefore P0 and more generally Pl As before we have the following L where dPi17Pl Notice how Ax is no longer the same on every subinterval We have Recall h ti m and gm As At a 07 and thus as n a 007 these becomes equality Now we have dltPi17 Therefore L Section 63 Instructor Rebecca Kalhorn Calculating Arc Length Let f be a function such that the derivative of f is continuous on the closed interval 17 The arc length of f from x a to z b is the de nite integral Let x ht and y gt be parametric functions such that the derivatives of h and g are continuous on the closed interval 7 s The arc length of the graph of this system of parametric functions from t r to t s is the de nite integral Note Because of the presence of a square root in each calculation of arc length7 it is often dif cult or impossible to evaluate it explicitly Therefore an approximation is often required 3 Example 1 Determine the length of y i 1 1 7 between 7 S x S 1 6 2m 2 Example 2 Determine the length of y 1 7 2 between 0 S x S 1 Section 63 Instructor Rebecca Kalhorn 2 Example 3 Determine the length of z 7132 between 1 S y S 4 Example 4 Determine the length of z 6t cost7 y it sint between 0 S t S 7139 Section 64 Instructor Rebecca Kalhorn Average Value of a Function Consider We want to nd the average value of a function on a given interval using Calculus Applications Daily temperature7 Velocity of a car or other automobile7 Electricity7 etc Recall If your test scores are 907 857 707 and 957 what is your average test score So7 calculating the average of nitely many numbers is an easy task How do we calculate the average value of a function for some interval7 or in nitely many numbers Suppose f is a continuous function on the interval 17 The average value of this function on 17 can be approximated by taking the As 71 gets larger the approximation will get better In fact as n a 00 the approximation approaches the Let Ax Divide 17 into n subintervals of length Ax and pick an m in each subinterval The average of the numbers fz1 is Therefore the average value of f on the interval 177 fam7 is fave Mean Value Theorem for Integrals If f is continuous on 177 then there exists a number 0 E 17 such that Equivale ntly7 Section 64 Instructor Rebecca Kalhorn Graphical Explanation Example 1 Find the average value of the function f 4 7 8x2 6 on the interval 073 Example 2 Find the average value of the function f x2 75x 6 cos7rx on 17 Section 64 Instructor Rebecca Kalhorn Example 3 Determine the number 0 that satis es the Mean Value Theorem for Inte grals for the function f x2 3x 2 on the interval 174 Example 4 In a certain city the temperature in degrees Fahrenheit t hours after 9 am was modeled by the function Tt 5014 sin Find the average temperature during the period from 9 am to 9 pm Section 65 Instructor Rebecca Kalhorn Applications to Physics and Engineering Applications Considered work7 force due to water pressure7 and centers of mass Strategy Break up the physical quantity into a large number of small parts7 as 0111 number of small parts add the results7 take the and evaluate the resulting Work a a De nition Work is the total amount of required to perform a task ln physics7 work has a technical meaning that depends on De nition Force describes the on an object Newton7s Second Law of Motion If an object moves along a straight line with position function 5t7 then the force F on the object in the same direction is de ned as the When acceleration is constant7 and the work done is the product of the and the W Hooke7s Law The force required to maintain a spring stretched x units beyond its natural length is to 7 ie fx 7 where 7 is a positive constant called the Note Hooke7s Law holds provided that z is not too large Units of Measure Internationalmetric System English System mass uid ounces ounces oz pounds lbs meters seconds s seconds s force weight ounces oz pounds lbs newton meter joule J Section 65 Instructor Rebecca Kalhorn Example 1 Suppose a spring has natural length 2 feet and that a force of 20 pounds is needed to compress the spring to a length of 18 inches Find the amount of work necessary to stretch the spring from a length of 25 feet to a length of 3 feet Example 2 Suppose a cylindrical tank has height 107 the radius of the base is 77 and it is half lled with water Find the amount of work necessary to move all the water out of the top of the tank Example 3 A cable that weighs 2 lbsft is used to lift 800 lbs of coal up a mineshaft 500 ft deep Find the work done Section 65 Instructor Rebecca Kalhorn Hydrostatic Pressure and Force De nition We de ne to be the force that is exerted on a vertical plate submerged in water due to the pressure of the water Two Basic Formulas 1 If we are d meters below the surface7 then is given by where p is the and g is the gravitational acceleration If we assume the uid in question is water then we have Note When dealing with English units the density measurement Assume that a constant pressure P is acting on a surface with area A Then7 A D acting on this area is We wont be able to nd the hydrostatic force on a vertical plate using only this formula since pressure will We will use calculus to remedy this situation Example 4 A swimming pool is 20 ft wide and 40 ft long and its bottom is an inclined plane7 the shallow end having a depth of 3 ft and the deep end7 9 ft If the pool is full of water7 nd the hydrostatic force on the shallow end and the deep end Instructor Rebecca Kalhorn Sectlon b 5 Moments and Centers at Mass De nition The pornt P on whrch a thrn plate ol any gwen shape balances hornontally all d e ol the plate A Suppose we have rmer many pornts mm amp wy wrth the mass at euh m m m then we can calculate the lollowrng pornt gwen y s The moment ol the system about the orrgrn wrth respect to the yams rs My s The moment ol the system about the orrgrn wrth respect to the wraxls rs Ma s The center ol mass ol the system rs ls the where M wlth unllorm de ned ln terms ol the area between two contrnuous lunctrons 3 m and 3 m wrth lor all a0 6 m1 Frrst we wlll spht our area up rnto rectangles by talnng lntexwls ol equal Consrder the lollowrng graph The pornts any represent the ol the rectangles so we can wrrte we we s The moment olT about the orrgrn wrth respect to the yams rs My Sectroh 6 5 Instructor Rebecca Kalhom A The moment ofT about the ongm wrth respect to the wax rs A The center of mass called the centroid ofT is the mass ofT So we can also wrrte the where M cehtrord as A Example 5 Fmd the exact coordinates of the cehtrcrd for the regrch bounded by the curvesye2yoahde2 A Example 6 Fmd the center of mass of the toucwrhg regrch Semen 6 5 Instructor Rebecca Kalhom A Example 7 Fmd the center of mass loe following region Section 6 5 Insuucioi Reheaca Kaihoin Applications to Physics and Engineeiing Woxksheet 1 A iank has ihe shape oi an inveiied cone Wiih heighi 10 m and base has a radius oi 4 m It is lled Wiih waiei to a heighi oi 2 m Find ihe Work iequiied Lo empiy ihe iank by H L L L v The densiiy 2 A veiiioai piaie is submerged in waiei and has ihe indicated shape Ekplajn how to ap pioxunaie ihe hydiosiaiio ioioe againsi one side oi ihe piaie by a Riemann sum Then express ihe ioioe as an iniegiai and ewluate IL 12 Calculus ll Notes Instructor Rebecea Burton Kalhorn Based off text Calculus Concepts and Contexts 3rd Ed by James Stewart Section 55 Instructor Rebecca Kalhorn The Substitution Rule Recall The chain rule is a powerful rule for differentiation It is used to nd the derivative of a function that is the composition of two or more functions For example hf9 ex h f 99 9 Consider Since integration is the inverse operation of differentiation we can use the above to come up with a rule for integration Suppose F f then we have f999 d96 F 99 99 d95 F995 C Now lets 77substitute77 u g to get fgzg xd c but c F udu fudu Substitution Rule If u g is a differentiable function whose range is an interval I and f is continuous on I7 then fltgltzgtgtg ltzgtdz fudu The challenges are knowing when to use substitution and making the appropriate substitution This can be done by asking a couple of questions ls there a function is nested in another function If so7 what is it ls a multiple of the derivative of that function present Practice will make nding the correct substitution second nature Example Find fsin3xdx Section 55 Instructor Rebecca Kalhorn Examples For each of the following use substitution to nd the inde nite integral Check your solutions by differentiating sin2x coszdx em3z2dz t idt 2 t24 Ld 1 2x35 Methods for Evaluating a De nite Integral Using Substitution 1 Evaluate the inde nite integral rst7 and then use the Fundamental Theorem of Calculus 2 If g is continuous on 17 and f is continuous on the range of u 9z7 then b 95 z x dx u du a f9 W gt a f gt I will use method 2 in class If you choose to use method 1 be sure you use correct notation Section 55 Instructor Rebecca Kalhorn Examples For each of the following use substitution to nd the de nite integral 7 a 39 V4 3xd 39 xv a2 7 zzds 0 0 1 1 lt Ch 3 0 1z2 cos7rtdt 1 Section 56 Instructor Rebecca Kalhorn Integration by Parts Recall We have seen that the Substitution Rule for integration correpsonds to the Chain Rule for differentiation Consider What would be the inverse operation of the Product Rule for differentiation Suppose u and 1 are differentiable functions with independent variable x From Calculus I we know d w 11 iu by the Product Rule Now we can use this to come up with a rule for integration 10111711 iu 7 11 iu i m 7 1 du u d1 dx dx Integration by Parts Let u and 1 be fucntions of x du denote the derivative of u7 and d1 denote the derivative of vthen udvuvivdu Tips for Deciding u and 1 d1 must be something you can integrate Choose the term more dif cult to integrate for u7 which hopefully makes d1 integrable In general you can use the acronym7 LIATE7 as a guide for choosing u L logarithm I inverse trig function A algebraic fucntion T trig function E exponential function Obviously the resulting intgral7 f1 du7 should not be more dif cult to integrate than the original7 fu d1 Section 56 Instructor Rebecca Kalhorn Examples z 1ndz 3 7 5x COS4dz 7 7 3gt66md zzcos5zdx Section 56 Instructor Rebecca Kalhorn tan 1dz tzetdt 39 Challenge Problem ew sinzdx Section 56 Instructor Rebecca Kalhorn Method for Evaluating a De nite Integral Using Integration by Parts b b b f969 96d96f96996 e 9xf xdx a Examples 1 tan 1 dx 0 141ntdt Appendix G Instructor Rebecca Kalhorn Integration of Rational Functions by Partial Fiactions Suppose we want to evaluate the following 4 7 1 95 6195 x2 z 7 2 If we knew that we could rewrite the integrand as 3 1 z2 zil then we could evaluate idx d How do we do this x 2 z 71 Recall A rational function is a function of the form where P and Q are polynomials If the degree of P is less than the degree of Q7 then f is called a proper rational function lfthe degree of P is greater than the degree of Q7 then f is called an improper rational function7 and f can be rewritten as where S and R are also polynomials Consider We can use common denominators in order to combine several rational functions into one rational function Now we want to reverse this process and rewrite rational functions as the sum of simplier rational functions7 called partial fractions Steps for Finding Partial Fraction Decomposition Pz i i l ls f z a proper rational function Qltzgt No Rm a Do long division to get f SW QW and procede to 1b Yes b ls Qz factorable Appendix G Instructor Rebecca Kalhorn No gt Can7t do partial fraction decomposition Yes Case I Qz is a product of distince linear factors Suppose Q alx b1a2x b2 mm bk7 then our partial fraction decomposition is A1 A2 Ak m In m b2 39 39 39 akz b Example Find the partial fraction decomposition of 20 2x273z7 239 Case II Qz is a product of linear factors7 where some repeat Suppose azb is a linear factor of Qz that is repeated k times7 we use A1 A2 Ak ax b cw b cw b as part of the partial fraction decomposition Every repeated lin ear factor is treated this way and each linear factor that appears only once is treated as in Case I Example Find the partial fraction decomposition of 1 4 i ff Appendix G Instructor Rebecca Kalhorn Case III Qz is a product of at least one distinct irreducible quadratic fac tor and possibly some linear factors also may be distinct andor non distinct Suppose axz bx c is a distinct irreducible quadratic factor of Qx7 we use Ax B ax bx c as part of the partial fraction decomposition Linear factors are treated as in Case I and II Example Find the partial fraction decomposition of 3 7 4x2 2 2 12 239 Note The partial fractions will be of the form A Ax B 7 or zzll ax bxcl7 where A7 Babc7 and 239 are numbers Example Set up the form for the partial fraction decomposition for the following I x2 2gtltz2 7 W 22z 7 2 Now we use the process of partial fraction decomposition in order to integrate rational func tions previously unintegrable Appendix G Instructor Rebecca Kalhorn Example Find the following intgrals 3 d 71 72x 14 dz x2 2x 7 3 6z2 35x 50 i 96 4296 7 2 Section 57 Instructor Rebecca Kalhorn Additional Techniques of Integration Trigonometric Integrals 39 We can use the substitution rule and integration by parts to help us with solving integrals whose integrand is a trigonometric function and those whose antiderivative is an inverse trigonometric function We can also use trigonometric identities and formulas to rewrite the integrand of certain integrals in order to integrate You Should Know I 17d tan 1x c sinzdx 7 cosx 0 dx sin 1x c coszdx sinz c dx cos 1x c sec2d tan c fl 172 2 2 7 39 cos zs1n 71 i Identities and Formulas Not to Memorize Double Angle Formulas Half Angle Formulas 39 mm 2W COS t 2 gt 7 39 003295 008295 i sin2z Sm 95 i 2 Qtanw 1 cos 2x tame m 39 cos2x New Integrals to Memorize 1 1 72 de itan71c z a 1 a Proof Section 57 Instructor Rebecca Kalhorn Reduction Formulas sin dz fl cosx sinn lw n 7171 sin 2d7 n 2 2 Proof 71 Cos zdx l cosn 1z sinx n 77 Cos 2dz7 n 2 2 The proof is similar to that for sine7s reduction formula Examples Evaluate each of the following 39 sin3dz Use two different methods Section 57 Instructor Rebecca Kalhorn 16sin4x cos4dz 39 tan3z secxdz Use the substitution u sec Section 57 Instructor Rebecca Kalhorn Trigonometric Integrals Often it is necessary to integrate functions which contain an expression of the form a2 7 2 a2 2 or 2 7 12 A trigonometric substitution is frequently useful in these situations ExampleUse integration to prove that the area of a circle is 7T7 2 where the radius is r Section 58 Instructor Rebecca Kalhorn Integration Using Tables and Computer Algebra Systems Table of Integrals Using a table of integrals can help to integrate functions that are dif cult to integrate by hand when a computer algebra system such as Maple is not available A brief table of integrals is included in the references pages at the back of the book a Note The Table of Integrals should be used as a tool not a cruch Examples Use the table of integrals to evaluate the following 2 2 z212dz 0 z 4 dz 5 7 4x2 Section 58 Instructor Rebecca Kalhorn Zsind x224dx Section 58 Instructor Rebecca Kalhorn Computer Algebra Systems While computer algebra systems are very useful in computing integrals7 hand compu tation is still important Often computers will return answers in a combersome form because of their matching process of solving integrals Examples 8 7dx 41 x224dx z2 58dz sin5zcos2dx Seotroh 5 9 Instructor Rebecca Kalhom Apploximation Integxation Recall A de mte mtegrsl rs equmleht to E r fdw rs equmleht to Sometrmes rt rs necessary to use approxrmatroh methods to estrmate the wlue of a de mte mtegal mm 0 Two of these srtuatrohs are A The function fw may not have In fact the majority of elementary functions do not have In other words rt rs common to be confronted wrth afuncmonA fwwh1ch we usmg basro mtegratroh formulas substrtutrom mtegratroh by pans table of Integrals or computer algebra systems A sl dt Noucethat w rs a oohtmuous uhotroh so butwe nu t We wru look at thrs more In Chapter 2 A few more such functions are Instructor Rehecca Kalhorh rim r r e r Wm A cos2 dw A A The luhctrch mt may Sectloh 5 9 In several applrcatrchs m rs based on collected data A Example A Example Recall We have already rhvestrgated apprcxrmatmg the de rute mtegral 1 through the use cl domam 39 d L l d lntelwl W In Calculusl rt was drscussed how to use a Rlemann sum wrth selected perms belng lelt endpom39s rrght endpoln39s or mrdpcmts cl the submtemls H We use a Riemann sum W39th mldpomts to estlmate the area A represented by mode we get Midpoint Rule then the Mrdpcmt Apprcxrmatrch cl 1 rs Sectron 5 9 Instructor Rebecca Kalhorn We can use trapezorde as opposed to rectangles for a more accurate approxrrnatron H we use trapezords to estrrnate t the area represented by mode we get Recall The calculatron of the area of a trapezord I Trapezord Area IYapezoidal Rule then the Trapezord Approxrrnatron of 1 re Sectloh 5 9 Instxuctox Rehem Kalhom ple Gwen the luhctloh f0 at 1 Approxlmate the true axea bounded by the graph at f the wraxls and the lmes 0 and a0 1 Use 5 Llapezolds mple Use the I lapezoldal Rule and the Mldpomt Rule Wlth 39n 5 to approxlmate A Ex the mtegxal 2 1 m Recall ll a graph at a luhctmh s the nghtsum Wlll be an t ble the leltsum approxlmatlon Wlll be an A d What and the Llapezoldalsum Here ls a a about the mlddlesum Lertsum Thapezoldal Sum Middlesum Rightsum Overestimzte Underestimzte Section 59 Instructor Rebecca Kalhorn We will introduce a third way to approximate integrals Simpson7s Rule Suppose n is the number of subintervals7 1727 an are the midpoints of the subintervals7 and 104117 an are the endpoints of the subintervals then the Simpson Approximation of 1 is 2 1 Example Use Simpson7s Rule with n 5 to approximate the integral idz 1 s W 39 We de ne the error 1 Midpoint Rule 6M 2 Trapezoid Rule 6T 3 Trapezoid Rule 65 Midpoint Rule Error Bound Suppose lf xl S K for all z E 17 then Trapezoidal Rule Error Bound Suppose lf xl S K for all z E 17 then Simpson7s Rule Error Bound Suppose lf4l S K for all z E 17 then Section 59 Instructor Rebecca Kalhorn More explanation on how we arrive at these error bounds is found on pages 414 415 and 419 of your book Example For what smallest value of 71 does the Trapezoidal Rule provide an approxi mation for the 6de such that leTl S 10 8 0 4 Example Maple found the Trapezoidal Rule approximation for 3 7 3dx with 0 n 10 to be T10 5264 Calculate the error bound Find the de nite integral Calculate the error Section 510 Instructor Rebecca Kalhorn Improper Integrals Recall We know how to evaluate de nite integrals of the form 1b fzd where a and b are and f is Consider Can we still evaluate the de nite integral if What if f is Improper Integrals Type I Type II Type I t Case I If fzd exists for every number t 2 17 then b Case II If fzd exists for every number t 3 b7 then if Type II Case I If f is continuous on ab and discontinuous at b7 then Case II If f is continuous on ab and discontinuous at 17 then Section 510 Instructor Rebecca Kalhorn De nition An improper integral is called if the corresponding limit exists and if the limit does not exist 39 lf fxd and fxd are convergent then c b If f has a discontinuity at c where a lt c lt b and both fzd and fzd are convergent then a c Examples For each of the following determine whether the integral is convergent or diver gent lf it is convergent then evaluate it 1 d 0 3x 1 95 Section 510 Instructor Rebecca Kalhorn 00 gd OO 33 71 15d 0 011nzd Section 510 Instructor Rebecca Kalhorn Important Result 00 ids is 17 1 Proof Comparison Theorem Suppose that f and g are continuous functions with f 2 g 2 0 for z 2 a If fzd is convergent7 then If gxdx is divergent7 then s2ltzgt co Example Use the Comparison Theorem to determine whether de is con 1 m vergent or divergent Section 510 Instructor Rebecca Kalhorn Improper Integrals Worksheet 1 6m 1 Evaluate idm 1 6m 7 1 2 zezmdx 00 1 3 Use the Comparison Theorem to determine convergence or divergence for dx 1 m e


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