1、Chapter 5 Integrals5.1 Areas and Distance5.2 The Definite Integral5.3 The Fundamental Theorem of Calculus5.4 Indefinite Integrals and the Net Change Theorem5.5 The Substitution Rule5.6 The Logarithm Defined as an Integral5.1 Areas and Distances The area problem 1.Areas of curved trapezoidSuppose the
2、 curved trapezoid is bounded by0),0)()(yxfxfyandbxax,Find the area of A.?A)(xfy Area of rectangleahhaahbArea of trapezoid)(2bah1xix1ixxabyoMethod:1)Partition:bxxxxxann1210,1iiixxUse the linesixx to divide A into small curved trapezoid;2)Approximation:,1iixxBase:)(ifHeight:We can approximatebyiA)()(1
3、iiiiiixxxxfA),2,1,niiSmall rectangle:3)Sum:niiAA1niiixf1)(4)Limit:Let,max1inixthen the area of the niiAA10limniiixf10)(limxabyo1xix1ixicurved trapezoid is:Riemann sum2.Distance,)(21TTCtvvand,0)(tvFind the total distance s.Method:1)Partition:,1iiitt1,isiitt2)Approximation:1use()to replace()over,iiivv
4、 tttWe get iiitvs)(),2,1(nisi),2,1(niSupposeSuppose the distance over212101TtttttTnn3)Sum:iniitvs1)(4)Limit:iniitvs10)(lim)max(1initCommon characteristic of the above problems is:1)The process is the same:“Partition,Approximation,Sum,Limit”2)The limit forms are the same:Both are the limits of Rieman
5、n sumsRiemann sum5.2 The definition of integralabxo The definition of integralSuppose()isdefinedon,.f xa bFor anypartition of ,a b,210bxxxxan1let,iiixxx,1iiixxi1aslongasmax0,ii nx iniixf1)(always tends to I,We say that I is the integral of)(xfover,a b1xix1ixbaxxfd)(That is baxxfd)(iniixf10)(limThen
6、we say that f(x)is integrable on a,b .denote it byniiixf1)(iniixf10)(limbaxxfd)(:where:integral signf(x):integranda :lower limit of integrationb :upper limit of integration The procedure of calculating an integral is called integration.Caution:The definite integral is a number;it does not depend on
7、x.In fact,we could use any letter in place of x without changing the value of the integral:badxxf)(bababadrrfdttfdxxf)()()(where x is a dummy variable.Geometric interpretation:Axxfxfbad)(,0)(:the area of the regionbaxxfxfd)(,0)(:the negative of the area of the regionabyx1A2A3A4A5A54321d)(AAAAAxxfba:
8、the algebraic sum of the oriented area,or the net area.ATheorem1.()is continuous on ,f xa bTheorem2.()is bounded on ,f xa band have only finite discontinuities on a,b The sufficient condition of integrability:)(xfis integrable on,ba)(xfis integrable on,bao1 xyniExample 1.Use the definition to evalua
9、te.d102xxSolution:Useniix),1,0(ninix1,niiWe choose),2,1(ni2xy 2then()iiiifxxto divide 0,1 into n subintervals of equal width.o1 xyniiinixf)(1niin1231)12)(1(6113nnnn)12)(11(61nniniixxx120102limdnlim31)12)(11(61nn2xy Example 2.Use the definition to evaluate.d31xexSolution:Useniix21),1,0(ninix2,12niiWe
10、 choose),2,1(nithen()iiiifxexto divide 1,3 into n subintervals of equal width.iinixf)(1ninien1/212ninixenxe1/210312limdnlimee 312/2/)2(/)23(nnnnneeen121lim)2(ppppnnnnnipn1lim1niiixninnin111lim)1(121lim)2(ppppnnnSolution:ninnin111lim)1(nninin11lim1iixxxd110 x01ni 1niUse the integral to denote the fol
11、lowing limits:xxpd10dxxRR022dxxRR022241R22xRyROUse the geometric meaning to evaluate:The Midpoint Rule1111()()()()whereandmidpoint of,2nbinaiiiiiif x dxf xxx f xf xbaxnxxxxx Properties of the definite integral When we defined the definite integral .we implicitly assumed that ab.Notice that if we rev
12、erse a and b,then change (b-a)/n from to (a-b)/n.Thereforebadxxf)(abbadxxfdxxf)()(If,then0 and soabx 0)(aadxxfProperties of the integral)(d.1abcxcbaxxfkxxfkbabad)(d)(.2(k is a constant)bababaxxgxxfxxgxfd)(d)(d)()(.3Proof:01The left sidelim()()niiiifgx0011lim()lim()the right sidenniiiiiifxgx(c is a c
13、onstant)Example:Use the properties of integrals to evaluate 102)34(dxx10210102101023434)34(:dxxdxdxxdxdxxSolution314410210dxxdxthatknowWebccabaxxfxxfxxfd)(d)(d)(.410880100)(12)(17)(:dxxffinddxxfanddxxfthatGivenExample5.()0,()0.6.()(),()().7.(),()()()babbaabaIf f xforaxb thenf x dxIf f xg x for axb t
14、henf x dxg x dxIf mf xM foraxb thenm baf x dxM baComparison Properties of the integral:The next theorem is called Mean Value Theorem for Definite Integrals.Its geometric interpretation is that,for a continuous positive f(x)on a,b,there is a number c in a,b such that the rectangle with base a,b and h
15、eight f(c)has the same area as the region under the graph of f(x)from a to b.oabxycf(x)Theorem If f(x)is continuous on a,b,then there exists at least a number c in(a,b)such that baabcfdxxf).)()(The number is called the average value of f(x)on a,b.Proof If f(x)is a constant function,the result is tru
16、e.Next we assume that f(x)is not a constant function.Since f(x)is continuous on a,b,f(x)takes on the minimum and the maximum values on a,b.Let f(u)=m and f(v)=M be the minimum and the maximum values of f(x)on a,b,respectively.Then m f(x)M for some x in a,b because f(x)is not constant.Therefore,we ha
17、ve It follows that ()/()baf x dx ba()bbbaaamdxf x dxMdx()()()bam baf x dxM ba The preceding inequalities indicate that the number is between m=f(u)and M=f(v).Thus the Intermediate Value Theorem leads that there is a number c between u and v such that Multiplying the both sides by b-a gives the concl
18、usion of the theorem.()/()baf x dx ba()()baf x dxf cba5.3 The Fundamental Theorem of Calculus The Fundamental Theorem of Calculus The fundamental Theorem of Calculus is appropriately named because it establishes a connection between the two branches to Calculus:differential calculus and integral cal
19、culus.Differential calculus arose from the tangent problem,whereas integral calculus arose from a seemingly unrelated problem,the area problem.The Fundamental Theorem of Calculus gives the precise relationship between the derivative and the integral.Newtons teacher at Cambridge,Issac Barrow discover
20、ed that the two problem are actually closely related.In fact,he realized that differentiation and integration are inverse processes.It was Newton and Leibniz who exploited this precise relationship and use it to develop calculus into a systematic mathematical method.In particular,they saw that the F
21、undamental Theorem enabled them to compute areas and integrals very easily without having to compute them as limits of sums.The first part of the Fundamental Theorem deals with functions defined by an equation of the form where f is a continuous function on a,b and x varies between a and b.Observe t
22、hat g depends only on x,which appears as the variable upper limit in the integral.If x is a fixed number,then the integral is a definite number.If we then let x vary,the number also varies and defines a function of x by g(x).xadttfxg)()(baxf(t)(xfy xbaoy)(xxhxThe Fundamental Theorem of Calculus,Part
23、 IIf(),thenf xC a bxattfxd)()(Proof:,bahxxthenhxhx)()(h1xahxattfttfd)(d)(hxxttfhd)(1)(f)(hxxhxhxh)()(lim0)(lim0fh)(xf)(xiscontinuous on ,anddifferentiable on(,)and()()a ba bxf x,)(baCxf2021)(1)(:1xxgthendttxgExamplex)2/sin()()2/sin()(2202xxSthendttxSExamplex.sec341xtdtdxdFindExampleuxtdtdxdtdtdxdthe
24、nxusetSolution114secsec:43414)sec(sec)()sec(xxdxduuRuleChainthebydxdutdtdudu?)(?)(?)()()()()(xhxgbxgxgadttfdxddttfdxddttfdxdThe Fundamental Theorem of Calculus,Part 2thenxffunctioncontinuousaoftiveantiderivaanisxFSuppose,)()()()(d)(aFbFxxfbaProof:According to part 1,)(d)(xfoftiveantiderivaanisxxfxas
25、oCxxfxFxad)()(Set,xaWe get(),CF aso)()(d)(aFxFxxfxaLet,xbwe have)()(d)(aFbFxxfbadenotebabaxFxF)()(315dxeExamplexeeex331|1024dxxExample313|103x636xdxExample3ln6lnln|63xbxdxExample0cos7bbxbsin0sinsinsin|011211211()21Butispositive!|dxxxx Why?Summary:We end this section by bringing together the two part
26、s of the Fundamental Theorem.,1)()(),()().2)()()(),xabaSuppose f is continuous ona bwe haveIf g xf t dt then g xf xf x dxF bF awhere F is an antiderivativeof f that is Ff The Fundamental Theorem of Calculus5.4 Indefinite Integrals and the Net Change Theorem Indefinite Integrals and the Net Change Th
27、eorem An indefinite integral of f(x)represent an entirely family of functions whose derivative is f(x)and is denoted by Suppose that F(x)is an antiderivative of f(x),then according to the theorem in 4.2,we know that any antiderivative G(x)of f(x)can be written as G(x)=F(x)+Cdxxf)(CxFdxxf)()(Then:Cau
28、tion:1)You should distinguish carefully between definite and indefinite integrals.A definite integral is a number,whereas an indefinite integral is a function(a family of functions).2)The connection is the fundamental theorem:babadxxfdxxf)()(xdd)1(xxfd)()(xfTable of Indefinite integrals From the abo
29、ve definition,we known that:dxxfd)(xxfd)(orCxd)2()(xF)(xForCd)(xF)(xFxkd)1(k is a constant)Cxk xx d)2(Cx111xxd)3(Cx ln0 x)1()ln()ln(xxx121d)4(xxCx arctanxxdcos)6(Cx sinxx2cosd)8(xxdsec2Cx tanorCx cotarc21d)5(xxCx arcsinorCx cosarcxxdsin)7(Cx cosxx2sind)9(xxdcsc2Cx cotxxxdtansec)10(Cx secxxxdcotcsc)1
30、1(Cxcscxexd)12(Cexxaxd)13(Caaxln2shxxeexCx chxxdch)15(Cx shxxdsh)14(2chxxeexExample 1:.d3xxxSulotion:=xxd34134Cx313Example 2:.dcossin22xxx Solution:=xxdsin21Cx cos21134xCThe properties of Indefinite Integralsxxfkd)(.1xxgxfd)()(.2Corollary:If,)()(1xfkxfiniithenxxfkxxfiniid)(d)(1xxfkd)(xxgxxfd)(d)()0(
31、kExample 3:.d)5(2xexx Solution:=xexxd)25)2()2ln()2(eex2ln25xCexx2ln512ln2CExample 4:.dtan2xxSolution:=xxd)1(sec2xxxddsec2Cxx tanExample 5:.d)1(122xxxxxSolution:=xxxxxd)1()1(22xxd112xxd1xarctanCx lnExample:.d124xxx Solution:=xxxd11)1(24xxxxd11)1)(1(222221dd)1(xxxxCxxxarctan313Application:The Net Chan
32、ge Theorem The integral of a rate of change is the net change:)()()(aFbFdxxfba)()()(aFbFdxxFba()()()()()()()()()()()()total distance traveledbabbaabbaabaC x dxC bC av t dts t dts bs aa t dtv t dtv bv av t dt Example 6 A particle moves along a line so that its velocity at time t is 1)Find the displac
33、ement of the particle during the time period 2)Find the distance traveled during the time period.6)(2tttv41 t29)()1()4()141dttvss17.10661)()241dttvSolution:5.5 The Substitution RuleThe Substitution Rulecxcuuduxxddxxxdxxu2sin21sin21cos21)2(2cos2122cos212cos2Example:)2(2xddx ceceduedxedxxedxxexuuxuxxx
34、2222221212121)2(212)(22xdxdx The Substitution Rule:CxFCuFduufxdxfdxxxfxu)()()()()()()()(fF Suppose()11(32)232dxxCx 23ln21dxx231Exampe 1 duuxu12123Cu ln21Example 2 dxxx212211(1)2x dx 3221(1)3xC 2211(1)2xxdx2112uxudu cu 2331Example 3dxxex332()xedx32(3)3xedxCex332323uxue du23uecExample 4)ln31(xxdxxxdln
35、31)ln31(31.|ln31|ln31Cx dxxxcossinxdxcoscos1Cx cosln xdxtanCxsecln xdxcotCxcscln xdxtanExample 5 ln secxC221d xax dxaxa22111 axdaxa21111arctanxCaa(1).dxxa221(2).arcsin xCa dxaxa2111 axdax211dxax221(3).Caxaxlna 211112dxaxaxa1()()2d xad xaaxaxaEx6322xxdx22)2()1(xdx22)2()1()1(xxd.21arctan21CxEx7dxxx249
36、1dxx2491dxxx24922)2(3)2(21xxd2249)49(81xxd32arcsin21x.49412Cx Definite Integrals The Substitution Rule for Definite IntegralsIf is continuous on a,b and f is continuous on the range of u=g(x),theng)()()()()(bgagbaduufdxxgxgf)()(|)()()(agFbgFxgFdxxgxgfbaba)()(|)()()()()()(agFbgFuFduufbgagbgagProof:Su
37、ppose F be an antiderivative of f.Then F(g(x)is an antiderivative of)()(xgxgfEx1.Evaluate).0(d022axxaaSolution:Let,sintax then,dcosdttax;0,0txAs.,2taxasThe above integral=2attad)2cos1(2202)2sin21(22tta0242a20ttdcos222xayxoyaSandEx2.Evaluate.d12240 xxxSolution:Let,12 xtthen,dd,212ttxtx,0 xAs,4xas.3tT
38、he above integral=ttttd231212ttd)3(21312)331(213tt 13322;1tand Ex3.,)(aaCxfSupposeProof:(1)If,)()(xfxfaaaxxfxxfthen0d)(2d)(xxfaad)(2)If,)()(xfxf0d)(aaxxfthenxxfad)(0 xxfad)(0ttfad)(0 xxfad)(0 xxfxfad)()(0,d)(20 xxfa)()(xfxfastxLet,0)()(xfxfasQuestion:?)(1)(226dxxfxxfSuppose?)()1/(tan)(If1142dxxfxxxx
39、f5.6 The Logarithm Defined as an Integral The Logarithm Defined as an Integral Our treatment of exponential and logarithmic functions until now has relied on our intuition,which is based on numerical and visual evidence.Here we use the Fundamental Theorem of Calculus to give an alternative treatment that provides a surer footing for these functions.01ln1xdttxxDefinition The natural logarithmic function is the function defined byBased on the above definition of natural logarithmic function,we will deduce all properties of logarithm functions。As far as the details,we left to you as an exercise.
