定積分的概念和性質(zhì)(Concept

1、第五章定積分Chapter 5 Definite Integrals5.1 定積分的概念和性質(zhì)( Concept of Definite Integral and its Properties )一、定積分問題舉例( Examples of Definite Integral )設在y = f x區(qū)間a,b 1上非負、連續(xù)，由x =a , x = b , y =0以及曲線y = f x 所圍成的圖形稱為曲邊梯形，其中曲線弧稱為曲邊。Let f x be continuous and nonnegative on the closed interval la,b .L Then the regi

2、on bounded by the graph of f x , the x -axis, the vertical lines x = a, and x = b is called the trapezoid with curved edge.黎曼和的定義(Definition of Riemann Sum)設f x是定義在閉區(qū)間l.a,b 1上的函數(shù)，二是l.a,b丨的任意一個分割，a =X0 : Xi ：|l： Xn4 :： Xn =b，其中是第i個小區(qū)間的長度，Ci是第i個小區(qū)間的任意一點，那么和nZ f (G )Xi , Xi4 蘭 c 蘭 Xii d稱為黎曼和。Let f x be

3、 defined on the closed interval La,b l, and let : be an arbitrary partition of l.a,b I,a = x()vXnvXn = b, where is the width of the i th subi nterval. If ci is any point in the i th sub in terval, the n the sumn' f Ci ，Xi , Xi 4 空 c 空 Xi,i TIs called a Riema nn sum for the partiti on :二、 定積分的定義(

4、Definition of Definite Integral )定義 定積分(Definite Integral)設函數(shù)f x在區(qū)間！a,b丨上有界，在a,bl中任意插入若干個分點a二滄：:Xi ： |1( : xn 4 :： Xn =b，把區(qū)間！a,b】分成n個小區(qū)間:Xo,Xi , l_Xi,X2 ,|, Xn,Xn 1,各個小區(qū)間的長度依次為LXjXj-Xo，二乂2=乂2-捲，LXnXn-Xnj。在每個小區(qū)間&丄x】上任取一點© ,作函數(shù)f（ ©）與小區(qū)間長度心 的乘積f G QX（i =1,2j|, n），并作出和nS = ' f 。i 4記=ma

5、Ax!x2J|jixn ，如果不論對a,b怎樣分法，也不論在小區(qū)間上點怎樣取法，只要當|P t 0時，和S總趨于確定的極限I，這時我們稱這個極限I為函數(shù)f x在區(qū)間l.a,b 1上的定積分（簡稱積分），記作:f x dx，即ba f X dx=l其中f x叫做被積函數(shù)，f x dx叫做被積表達式，x叫做積分變量，a叫做積分下限， b叫做積分上限，|a,b I叫做積分區(qū)間。Let f (x ) be a function that is defined on the closed intervala, b】.Consider a partition p of the in terval b,b

6、J into n sub in terval (not n ecessarily of equal len gth ) by means ofpoints a = x0 ： % ： 11 ( ： xn：xn = bandlet * 二 Xj -Xj.Oneachsub in terval Xj,x】,pick an arbitrary pointi (whichmay be an end point );wecall it asample point for the ith sub in terval.We callthe sumnS 八 f -Xj a Riemanni =1sum forf

7、 Xcorresponding to the partitionp.nIf Pm f(勺禺 exists,1 iwesay f X isintegrableon a,b 丨,wherep =max' % X2,|",. Moreover,bf X dx ,called definite integral (or RiemannaIntegral) off X from a to b ,is given bya f XdX=ipm f i *nThe equality二 > 0 such that”卩' f - -X = L means that, corresp

8、onding to each : >0,there is annZ f (勺 X L v 名 for all Riemann sums Z f (畤崔Xjfor f (X )i 1ion ta, b 】for which the norm P of the associated partition is less than 6 .bIn the symbol g f X dX, a is called the lower limit of integral , b the upper limit of integral,and la,b J the integralinterval.定理

9、 1 可積性定理 (Integrability Theorem )設f x在區(qū)間la,b I上連續(xù)，則f x在l.a,b 上可積。Theorem 1 If a function f X is continuous on the closed interval I a, b 1 ,it is in tegrable on a, bl.定理 2 可積性定理(Integrability Theorem )設f x在區(qū)間la,b 上有界，且只有有限個間斷點，則f x在區(qū)間la,b 1上可積。Theorem 2 If f x is bounded on la, b l and if it is cont

10、inuous there except at a finite number of points ,then f x is integrable on a,b l三. 定積分的性質(zhì)(Properties of Definite Integrals )兩個特殊的定積分a(1) 如果f x在x = a點有意義，則f x dx = 0 ；La(2) 如果f x在l.a,b 1上可積，則f x dxf x dx。L baTwo Special Definite Integralsa(1) If f x is defined atx=a.Then f x dx = 0 .a If f x isin te

11、grable onla, b 1. The n bfxdx- f x dx .定積分的線性性(Linearity of the Definite Integral )設函數(shù)f x和g x在la,b 1上都可積，k是常數(shù)，則kf x和f x + g x都可積， 并且bb(1) a kf x dx = k a f x dx；bbb a f (x )+g (x fldx = f (x px + L g (x px ； and consequently,bbb(3) a |f x -g x dx = a f x dx- a g x dx.Suppose that f x and g x are int

12、egrable on a,bl and k is a constant . Thenkf x and f x g x are integrable ,andbb(1) a kf x dx= k a f x dx ;b _bb(2) a f (x )+g(x )dx=f (x px+ Ja g (x px; and consequently,b_bb(3) a Lf x -g x dx = a f X dx- a g X dx.性質(zhì)3 定積分對于積分區(qū)間的可加性(Interval Additive Property of DefiniteIn tegrals)設f x在區(qū)間上可積，且 a，b和c

13、都是區(qū)間內(nèi)的點，則不論 a，b和c的相對位置cbc如何，都有.f x dx = . f x dx + . f x dx。aabProperty 3 If f x is integrable on the three closed intervals determined by a， b， and c ,thencbca f xdx= a f x dx+ b f xdxno matter what the order of a， b，禾口 c.性質(zhì) 4如果在區(qū)間la,b】上f(x)h，則fldx= fdx = b-a。"-a aProperty 4 If f x =1 for ever

14、y x in a, b】,thenbb1dx = dx = ba.a- a性質(zhì) 5如果在區(qū)間l.a,b 1上 f x - 0,則f x dx _ 0 a b。aProperty5If f x is integrable and nonnegative on the closed intervalLa,b l,thenba f x dx _ 0 a ： b .推論1。2定積分的可比性(Comparis on Property for Defin iteIn tegrals)如果在區(qū)間a,b 1上，f x < g x，則bba f xdxE ag xdx,bbf (x)dx 氣 f(xjdx

15、。用通俗明了的話說，就是定積分保持不等號。Corollary 1, 2 If f x andg X) is integrable on the closed interval fa,b】,andba f xdx <bag xdxf x - g x for all x in la,b I.Then andJ f(x)dx 蘭f(xa十*aIn in formal but descriptive Ian guage ,we say that thedefi nitein tegralpreservesin equalities.性質(zhì) 6 積分的有界性(Boundedness Property

16、 for Definite Integrals )如果f x在la,b 上連續(xù)，且對任意的x ：二la,bl，都有 m乞f x - M，則bm b - a f x dx_ M b aaProperty 6 If f x is continuous ona,b】 and m_ f x - M for all x ina, b l.Thenbmb- a fxdx_Mb-a 。a性質(zhì) 7 積分中值定理(Mean Value Theorem for DefiniteIntegrals )如果函數(shù)f x在閉區(qū)間la,b 1上連續(xù)，則在積分區(qū)間 a, b L 上至少存在一點，使下式成立ba f xdx=f

17、ba，且1bf=b-a af xdx稱為函數(shù)f x在區(qū)間la,b 上的平均值。Property 7If f x is continuous ona,bl,there is at least one numberbetweena and b such thatba f xdx=fb-a，and1bf=b-a af xdxis called the average value of f xon l.a,b I.5.2 微積分基本定理(Fundamental Theorem of Calculus)一.積分上限的函數(shù)及其導數(shù)( Accumulation Function and Its Deriva

18、tive )定理 1 微積分基本定理 (Fu ndame ntal Theorem of Calculus)如果函數(shù)f (x )在區(qū)間Ia,b】上連續(xù)，則積分上限函數(shù)© (x)=f(tpt在Ia,b上可導，并且它的導數(shù)是xd J f (t dt'x = a= f x a 空 x 乞 b .dxTheorem 1 Let f x be continuous on the closed interval a,bl and let x be a (variable)poi ntin a,b .Thenaf tdtisdiffere ntiableonxr N d Ja f (t d

19、tla,b l,and dx定理 2 原函數(shù)存在定理(The Existenee Theorem of Antiderivative)x如果函數(shù)f x在區(qū)間a,b 1上連續(xù)，則函數(shù) x = f t dt就是f X在l.a,b 1上的一個原函數(shù)l-a, b 1 ,thenx = . f t dt isa'aTheorem 2 If f x is continuous on the closed interval an antiderivative of f x on 'a, b).牛頓-萊布尼茨公式(Newton-Leibniz Formula)定理 3 微積分第一基本定理 (f

20、irst Fun dame ntal Theorem of Calculus)如果函數(shù)F x是連續(xù)函數(shù)f x在區(qū)間la,b 上的一個原函數(shù)，則ba f xdx=F b F a稱上面的公式為牛頓-萊布尼茨公式.Theorem 3 Let f x be continuous(henee integrable ) on La, b ),and let F x be any antiderivative off x on l.a, b I .Thenba f x dx=F b - F awhich is called the Newton-Leibniz Formula .5.3 定積分的換元法和分部

21、積分法(integration by Substitution and Definite Intgrals byParts)一. 定積分的換元法 (Substitution Rule for Definite Integrals)二. 定理定積分的換元法(Substitution Rule for Definite Integrals)假設函數(shù)f x在區(qū)間la,b 上連續(xù)，函數(shù)x二-t滿足條件十：a," ： b；(2) t在匕/ 1(或丨)上具有連續(xù)導數(shù)，且其值域R廣a,bl,則有b:a f xdx f t ' t dt上面的公式叫做定積分的換元公式Theorem Let t

22、 have a continuous derivative on上，-I(oJ < I), and let f xbe continuous on B,b】.if) = a , % P )=b and the range of $ (x) is a subset ofLa,b l.ThenbI-''f(xdx=L P(t)F'(t)dt ,which is called the substitution rule for definite integrals.二.定積分的分部積分法 (Definite Integration by Parts)根據(jù)不定積分的分部積

23、分法，有bau XV' X dX =bu x v' x dx a-ilu x v x - u' x v x dxb- av x U' X dX簡寫為或b i b uv'dx=uv， vu dx aa abudv= Iauvvdu.According to the indefinite integration by parts ,bbJau(xv'(x )dx=Ju( X)v'( x)dxauxvx u'xvxdxb-f v(x )u'(x jdxavvFor simplicity ,bbb.uv'dx=uv vu&

24、#39;dxaaaorbbudv= 1- uv ' "aavdu.5.4 反常積分(Improper Integrals).無窮限的反常積分 (Improper Integrals with Infinite Limits of integration )t定義1設函數(shù)f x在區(qū)間|a, :上連續(xù)，取t a ,如果極限im_ a f x dx存在為有限值，則此極限為函數(shù) f x在無窮區(qū)間 a,上的反常積分，記作a： f x dx,即a f XdXXtlim f x dx exists andt門：aa,亠 | ,which is denoteda，XdXmWe say tha

25、t the corresp onding improper in tegraltaf XdX,convergesOtherwise ,the integral is siad to diverge.這時也稱反常積分 a "f x dx收斂；如果上述極限不存在，函數(shù)f x在無窮區(qū)間反常積分就沒有意義，習慣上稱為反常積分 J'f x dx發(fā)散.Let f x be continuous on |a,亠 j ,and t a .If the limit have finite value , the value is the improper integral of f x on

26、by f (x dx ,that is ,設函數(shù)f (x )在區(qū)間(- ,b 上連續(xù)，取tvb，如果極限Jimff(xdx存在且為有限值, 則此極限為函數(shù)f X在無窮區(qū)間_：,b 1上的反常積分，記作：_f X dx，即這時也稱反常積分bb_：f xdx=tlim .tf x dx，bf x dx收斂；如果上述極限不存在，就稱反常積分OQb_f X dx發(fā)散。-0Let f x be continuous on -:,b 1，andt : b.lf the limitbtlim -t f xdx exists and havefinite value, the value is the imp

27、roper integral of f x onis denoted byb! f x dx ,that is ,bb J xdx = tlim： .t f xdx，We say that the said to diverge.corresp ondingimproperin tegralcon verges. Otherwise,the integral is0 :定義 設函數(shù)f x在區(qū)間上連續(xù)，如果反常積分-f x dx和.f x dx都收斂，則稱上述反常積分 之和為函數(shù)f X在無窮區(qū)間 -二，二 上的反常積分，記作beoriJ xdx，即: 0 :f xdx=：f xdx+ 0 f X

28、dx0t=lim f x dx+ lim t . tt0這時也稱反常積分：f x dx收斂；否則就稱反常積分5f x dx：'f x dx 發(fā)散。Let f x be continuous on -：,亠 i.If both-bothe nf x dx is said to con verge and have: 0 :f xdxf Xdx+ 0 f xdx0t專m t f xdx + tlim.0f xdx.ow：；乂 f ( x px and 0 f (x px con verge,value、無界函數(shù)的反常積分(ImproperIntegra ls of Infinite Int

29、egrands)定義無界函數(shù)反常積分 (Improper Integra ls of Infinite Integrand)設函數(shù)f (x)在半開閉區(qū)間l-a,b上連續(xù)，且lim f(x) =：,f 一則b.tf(x)dx = limf (x)dxat b_a如果等式右邊的極限存在且為有限值，此時稱反常積分收斂,否則稱反常積分發(fā)散DeintionLet f (x) be continuous on the half-open intervala,b and supposethat lim f(x)二：.Thent fbtf (x)dx = lim f (x)dx at b_aProvided that this limit exists and is finite,in which case we say that the integralcon verge.Otherwise,we say that the in tegral diverges.無界函數(shù)的反常積分(ImproperIntegra Is o