How can we approximate the area of a region in the plane? What is an interpretation of the area under the graph of a velocity function?


 Lesley Osborne
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1 Mth 125 Summry Here re some thoughts I ws hving while considering wht to put on the first midterm. The core of your studying should be the ssigned homework problems: mke sure you relly understnd those well before moving on to other things (like old midterms) Antiderivtives You should know wht it mens for f(x) to be n ntiderivtive of g(x). Given two functions, f(x) nd g(x), you should be ble to sy whether or not f(x) is n ntiderivtive of g(x). How mny ntiderivtive does function hve? Wht is tht +C business ll bout? Ares nd Distnces How cn we pproximte the re of region in the plne? Wht is n interprettion of the re under the grph of velocity function? The Definite Integrl You should understnd the definition of the definite integrl nd its reltion to re under curve. You should be ble to use the midpoint rule to pproximte definite integrl. Problems re prticulrly nice The Fundmentl Theorem of Clculus Prt 1: If f is continuous on [, b], then g(x) = x f(t) dt is continuous on [, b] nd differtible on (, b), g (x) = f(x). Prt 2: If f is continuous on [, b], then b where F is ny ntiderivtive of f. Be sure you cn differentite functions like f(x) dx = F(b) F() g(x) = x 3 sinx e t2 dt using the chin rule nd prt 1 of the FTOC (see, e.g., problems 5052).
2 5.4  Indefinite Integrls nd the Net Chnge Theorem Here we get the nottion tht f(x) dx stnds for the most generl ntiderivte of f The Substitution Rule The substitution rule is the most importnt nd powerful tool for finding ntiderivtives. It cn be considered, to certin extent, the reverse of the chin rule for differentition. Substitution is wy of getting from one indefinite integrl to nother. When trying to find ntiderivtives, we my need to try severl different substitutions until hitting on one tht improves the integrl we re working with to the point tht we cn find the ntiderivtive. Sometimes, more thn one substitution, used in sequence, is n effective wy to go. Prctice will improve your bility to see the right substitutions. As we get more techniques for finding ntiderivtives, the substitution method will lwys be with us. It will py to mke sure you cn use the method well now. There re tons of prctice problems in this section to work on to improve your substitution bility (e.g., problems 744, nd 4970) Ares between Curves The first of our mny pplictions of the integrl is to find the re between curves. Are is the integrl of width In mny instnces you will wnt to express the re s n integrl in y rther thn in x. It very often helps to hve decent sketch of the region whose re you re trying to find. Note tht b b (f(x) g(x)) dx = (g(x) f(x)) dx so if your nswer comes out negtive (which is impossible for n re) check tht you hven t got the difference of the two functions in the wrong order Volumes (by Cross Section) Here is developed the ide tht volume is the integrl of (cross sectionl) re.
3 Although mny of the exmples we looked t involve solids of revolution whose crosssections re circles, this method pplies to ny solid tht hs cross sections whose re cn be expressed s function of x (or y) Volume by Cylindricl Shells The method of wshers/disks is gret, but in certin cses we cn result in n integrl we re unble to evlute, or, indeed, to setup. So we hve nother method: the method of cylindricl shells. Even if the wsher/disk method works, the cylindricl shells method cn be esier. Prctice will help you decide which method to use on given solid. Problems 714 on pge 469 re good for this Work Mny problems finding the work required to perform certin tsk cn be solved by cutting the tsk into pieces, pproximting the work to perform ech piece, summing these pproximtions, nd tking limit s the number of pieces goes to infinity. The result is, of course, n integrl. There re bsiclly three ctegories of problems you might see in this course: cble problems, spring problems, nd pumping/digging problems. You should know how to del with ll of them Averge Vlue of Function This is very short section. You should understnd the definition of the verge vlue of function on n intervl Integrtion by Prts You should understnd how to pply the integrtion by prts technique: u dv = uv v du You should be ble to recognize integrnds for which this technique is prticulrly pproprite, such s: 1. positive integer power of x times sin x, cosx, e x (or relted functions) 2. ln x times ny power of x 3. e x times sin x or cos x Integrtion by prts cn be the method of lst resort: if nothing else works, you cn lmost lwys try prts : if you cn differentite the integrnd, you cn use it. Whether it helps or not is nother question, but it works well with such function s ln x nd rcsin x.
4 7.2  Trigonometric Integrls You should be ble to integrte integrls of the form sin n x cos m xdx nd tn n x sec m xdx The strtegies for these integrls depend on the prities of m nd n (tht is, whether they re even or odd). You should prctice ll cses. Note tht the cse where the power of tn x is even nd the power of sec x is odd does not hve cler strtegy, so vriety of techniques my be needed Trigonometric Substitution This technique exploits trigonometric Pythgoren identities to give useful substitutions tht convert qudrtic expressions into squres of trig functions. This mkes it prticulrly useful for eliminting squre roots (i.e., when you see qudrtic expression inside squre root, there s good chnce tht trig substitution might be useful). One thing tht mkes this technique more work thn simple substitution is the work required to convert bck to the originl vrible once n ntiderivtive is found. One efficient wy to do this is the tringle method s described in lecture nd the text. Be sure to prctice this spect of this technique Prtil Frctions This technique is pplicble to integrnds tht re rtionl functions. You should know how to pply this technique to ny rtionl function with denomintor tht is fctorble s product of liner nd distinct qudrtic fctors The ide behind this method is completely lgebric: rtionl function whose denomintor is product of liner fctors cn be expressed s the sum of simpler rtionl functions, ech of which hs denomintor which is liner or qudrtic, nd constnt or liner numertor. Such simpler functions re esily integrted. Long division of polynomils my be necessry first step when the numertor hs degree equl to or greter thn the degree of the denomintor Strtegies for Integrtion This section is perhps the most importnt. On the exm, the integrls will simply be presented to you for you to solve. You will hve to decide which technique, or techniques, to pply to find the ntiderivtive.
5 There re 81 problems in this section, nd they re pretty much ll good prctice Numericl Integrtion When we wnt to evlute definite integrl, nd we cnnot find n ntiderivtive of the integrnd, we cn pproximte the vlue of the integrl. Three techniques we ve seen for doing this re Midpoint Rule Trpezoid Rule Simpson s Rule All three methods re quite similr, nd require us to smple the function t number of eqully spced vlues of x, then combine these vlue ccording to certin formul which depends on the method. Note the two distinct uses of these methods: pplied to n integrl of n explicitly given function for which we cnnot find n ntiderivtive, nd to n integrl of function given only through tble of vlues (such s exmple 5 in the text). The implementtion of the method is identicl in both cses Improper Integrls There re two types of improper integrls: 1. Those of the form f(x) dx, f(x) dx, or f(x) dx 2. Those of the form b f(x) dx where f(x) is discontinuous somewhere on the intervl [, b]. We tret improper integrls by defining them s the limits of proper integrls. For instnce, k f(x) dx = lim f(x) dx k Be sure you remember l Hospitl s rule, since it is often the tool needed to evlute the limits resulting from improper integrls Arc Length We cn express the rc length S of grph y = f(x) over the intervl x b vi n integrl: b S = 1 + (f (x)) 2 dx
6 For the vst mjority of functions f we encounter, the bove integrl will not be on we cn evlute exctly, since we will be unble to find the ntiderivtive of the integrnd. So, rc length problems on exms tend to be of two types: either they involve cleverly chosen function for which the integrnd is ntidifferentible, or you will be sked to pproximte the rc length (using, e.g., Simpson s rule). Center of Mss You should be ble to determine the center of mss (or centroid) of given plnr region. Differentil Equtions You should know wht it mens for function or eqution to be solution to given differentil eqution. You should be ble to verify whether or not given function of eqution is solution to given differentil eqution. You should be ble to solve mny seprble differentil equtions. A differentil eqution is seprble if it is of the form dy dx = f(y)g(x) nd cn be potentilly solved vi intergrtion: dy f(y) = g(x)dx. You should be ble to solve n initil vlue problem consisting of seprble differentil eqution nd n initil vlue, or point on the solution curve. You should be ble to setup nd solve vriety of ppliction problems. These will involve description of reltionship between two quntities (usully, quntity nd time) from which you will need to crete nd solve differentil eqution representtion. Some ctegories of this sort of problem include: popultion modeling, with both exponentil nd logistic models mixing tnk problems orthogonl fmilies of curves
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