定积分的概念和性质Concept.docx

1、定积分的概念和性质Concept第五章定积分Chapter 5 Definite Integrals5.1定积分的概念和性质( Concept of Definite Integral and its Properties )一、 定积分问题举例( Examples of Definite Integral )设在y = f x区间a,b 1上非负、连续，由x = a , x=b , y =0以及曲线y二f x所围成的图形称为曲边梯形，其中曲线弧称为曲边。Let f x be continuous and nonnegative on the closed interval a,bL Then

2、 the region bounded by the graph of f x , the x -axis, the vertical linesx 二 a, and x = b is called the trapezoid with curved edge.黎曼和的定义(Definition of Riemann Sum)设f x是定义在闭区间l.a,b 1上的函数，厶是l.a,b 1的任意一个分割，a =冷：Xi ： | | :人：xn = b，其中Ax是第i个小区间的长度，G是第i个小区间的任意一点，那么和nZ f (Cj)Ax， xiJL c 兰洛i V称为黎曼和。Let f x b

3、e defined on the closed interval !a,b l, and let : be an arbitrary partition of l.a,b I,a =怡： III ex*vxn =b, where Ax is the width of the i th subinterval. If ci is any point in the i th sub in terval, the n the sumnJ f ( c Hxi ， xi xi ,i TIs called a Riema nn sum for the partiti on二、 定积分的定义( Defini

4、tion of Definite Integral )定义 定积分(Definite Integral)设函数f x在区间！a,b丨上有界，在a,b丨中任意插入若干个分点 a =怡：:为：川：:：人4 : Xn =b，把区间 a,b 1 分成n个小区间：仪0必 1, I.x1,x2 1JH, l-xn4,xn ,各个小区间的长度依次为 二咅=%-乂0，二屜=灭2-為，XnXn-xn/。在每个小区间IXx 上任取一点 ,作函数f （匕）与小区间长度也x的乘积f（ 件（i =1,2,|i|,n ）,并作出和记P = max&2,川，也xn，如果不论对a,b怎样分法，也不论在小区间txxj上点怎样取

5、法，只要当 P t 0时，和S总趋于确定的极限I，这时我们称这个极限I为 函数f x在区间l.a,b 1上的定积分（简称积分），记作:f x dx，即na f xdx=l = IPm f i其中f x叫做被积函数，f x dx叫做被积表达式，x叫做积分变量，a叫做积分下限，b叫做积分上限，| a, b 叫做积分区间。Let f x be a function that is defined on the closed interval la, b .Consider a partitionp of the in terval b,binto n sub in terval (not n ece

6、ssarily of equal len gth ) by means ofpoints =x0 :为：川：:：xnxn = bf x corresponding to the partitionnIf f(勺岁x exists,Integral) of f x from a to b ,is given bynf xdx=ipmon nd 0 such that 瓦 f ( yiXj -L for all Riemann sums 瓦 f ( yiXj for f (x)i =1 i =1on a, b for which the norm P of the associated parti

7、tion is less than d .bIn the symbol i 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 1 可积性定理 (Integrability Theorem )设f x在区间la,b I上连续，则f x在la,b 上可积。Theorem 1 If a function f x is continuous on the closed interval I a, b 1 ,it is int

8、egrable on I a, b I.定理 2 可积性定理(Integrability Theorem )设f x在区间la,b 上有界，且只有有限个间断点，则 f x在区间la,b 上可积。it is con ti nu ous there exceptTheorem 2 If f x is bounded on la, b 丨 and if at a finite number of points ,then f x is integrable on a,b .L3.定积分的性质(Properties of Definite Integrals ) 两个特殊的定积分a(1)如果f x在x

9、=a点有意义，则.f x dx = 0；a(2)如果 f x 在 l.a,b 1 上可积，则 f x dx 二- f x dx。 b aTwo Special Definite Integralsa(1) If f x is defined atx=a.Then . f x dx = 0.a(2) If f x is integrable on !a,bL Then f x dx = f x dx.b a定积分的线性性(Linearity of the Definite Integral )设函数f x和g x在la,b 1上都可积，k是常数，则kf x和f x + g x都可积,并且b b(

10、1) kf x dx= k f x dx;a ab b b| f x 亠g x dx= f x dx+ g x dx; and consequently,b b bg x are integrable on a,b l and k is a constant . Thenb b(1)a kf x dx= k a f x dx;b b b| f xg x dx= f x dx+ g xdx； and con seque ntly,b b ba f X -g X dx= a f Xdx- ag xdx.性质3 定积分对于积分区间的可加性( Interval Additive Property of

11、 DefiniteIn tegrals)设f x在区间上可积，且 a , b和c都是区间内的点，则不论 a , b和c的相对位置c b c如何，都有.f xdx=. f x dx+. f x dx。a a b vProperty 3 If f x is integrable on the three closed intervals determined by a，b， and c ,thenc b ca f X dx= a f X dx+ b f x dxno matter what the order of a， b，禾廿 c.性质 4 如果在区间！a,b 1上 f x 三1，贝U 1dx

12、= dx = b-a。“ = aProperty 4 If f x _1 for every x in a, bl,thenb b1dx= dx = b - a .a a性质 5如果在区间l.a, b 1上 f x _ 0,则 f x dx丄0 a b 。Property 5 If f X is integrable and nonnegative on the closed intervalla,b l,thenbf x dx - 0 a b .a推论 1。2 定积分的可比性(Comparison Property for Definite Integrals)如果在区间l.a,b 1上,

13、f x岂g x，贝Ub ba f xdx E ag xdx，L f(x)dx 兰|f(x0x。用通俗明了的话说，就是定积分保持不等号。Corollary 1, 2 If f x andg X)is integrable on the closed interval fa,b】,andf x _g x for all x in l.a,b l.Thenandf (x)dx 兰 jjf (x jdx。In in formal but descriptive Ian guage ,we say that the defi nite in tegral preservesin equalities.性

14、质 6 积分的有界性(Boundedness Property for Definite Integrals )如果f x在la,b 上连续，且对任意的l.a,b 1，都有 m 乞 f x M，则bm b - a I f x dx辽 MProperty6 If f x is continuous on l.a,b 1and m _ f x - M for all x inl.a, b l.Thenbm b - a j I f x dx 乞 M b - a。性质 7积分中值定理(Mean Value Theorem for Definite Integrals )如果函数f x在闭区间la,b

15、1上连续，则在积分区间a,b 1上至少存在一点，使下式成立ba f xdx= f b-a，1 b f =L a f xdx b _a a称为函数f x在区间la,b 1上的平均值。Property 7 If f x is continuous on a, b,there is at least one number betweena and b such thatL f (x)dx= f 仁)(b-a),aandba f XdXis called the average value of f x on a,b.5.2微积分基本定理(Fundamental Theorem of Calculus

16、)一.积分上限的函数及其导数( Accumulation Function and Its Derivative )定理 1 微积分基本定理 (Fu ndame ntal Theorem of Calculus)X如果函数f X在区间la,b吐连续，则积分上限函数 X f t dt在la,b 上可导，并a v且它的导数是Xd f f (t dtx = = f x a 乞 x 乞 b .Theorem 1 Let f x be continuous on the closed interval a,bl and let x be a (variable)Xpoint in a, b .Then

17、X = ! f t dt is differentiable on a vxd J f (t dtla,b l,anda = f x a Ex Eb .dx定理 2 原函数存在定理(The Existenee Theorem of Antiderivative)如果函数f x在区间l.a,b 1上连续，则函数 x = Xf t dt就是f x在l.a,b 1上的一个原函数1 1 a v vxTheorem 2 If f x is continuous on the closed interval a,bthen X is1 1 aan antiderivative of f x on la,

18、b I.二.牛顿-莱布尼茨公式(Newton-Leibniz Formula)定理 3 微积分第一基本定理 (first Fun dame ntal Theorem of Calculus)如果函数F x是连续函数f x在区间la,b 1上的一个原函数，则bfxdx=Fb-Faa称上面的公式为 牛顿-莱布尼茨公式Theorem 3 Let f X be continuous(henee integrable ) ona,bl,and let F X be any antiderivative of f X on La, b).Thenba f xdx=F b -F awhich is call

19、ed the Newton-Leibniz Formula5.3定积分的换元法和分部积分法 (integration by Substitution and Definite Intgrals byParts)1.定积分的换元法 (Substitution Rule for Definite Integrals)2. 定理 定积分的换元法(Substitution Rule for Definite Integrals)假设函数f x在区间la,b 上连续，函数x二t满足条件讣：a,b; t在L：J：(或：，： I)上具有连续导数，且其值域R厂la,bl,则有b :a f XdX= f t t

20、 dt ,上面的公式叫做定积分的换元公式Theorem Let t have a continuous derivative on 上，-l(or-厂 I), and let f xbe continuous on la,bI .If : =a , ： =b and the range of x is a subset ofl.a,b l.Thenb :a f xdx= ： f t t dt which is called the substitution rule for definite integrals.二.定积分的分部积分法 (Definite Integration by Part

21、s)根据不定积分的分部积分法，有二 u x v x uxvxdx_ b-| u x v x ab-f v(x )u( x Jdx简写为b b buvdx=uv， vudxa a ab budv= I uv I - vdu .a a -According to the indefinite integration by parts ,fu(xy(x加ju x)vY x)dx=u( x)v( x)- Ju( x)v( x)dx=u x)v( x)b b- av x u xdxFor simplicity ,oruvdx= I uv 1ab-vudxab budv= I uv -a avdu.5.4

22、 反常积分(Improper Integrals).无穷限的反常积分 (Improper Integrals with Infinite Limits of integration )tf x在区间a,- :上连续，取t a ,如果极限im_ f x dx存在且定义1 设函数为有限值，则此极限为函数f x在无穷区间 a, v 上的反常积分，记作 f x dx,即a: ta f xdx=tlim af Xdx.这时也称反常积分 f x dx收敛；如果上述极限不存在，函数f x在无穷区间 a, 上的La -反常积分就没有意义，习惯上称为反常积分 f x dx发散.La. tLet f x be c

23、ontinuous on |a, ： ,and t a .If the limit pm一. f x dxexists andhave finite value , the value is the improper integral of f x on 2,亠 i ,whichis deno ted-beby f x dx ,that is ,a: tf x dx= lim f x dx,a t ： . aWe say that the corresponding improper integral convergesOtherwise ,the integraldiverge.is sia

24、d to b _ 一设函数f (x j在区间(-o,b 上连续，取tcb，如果极限lim f(xpx存在且为有限值,则此极限为函数f x在无穷区间-：，b 1上的反常积分，记作b f x dx，即b bf xdx=tlim t f xdx，这时也称反常积分bf x dx收敛；如果上述极限不存在，就称反常积分OQbf x dx发散。 bLet f x be continuous on andt : b.If the limit lim t f x dxexists and havefinite value, the value is the improper integral of f x on

25、 一匚亠b|,whichis denoted bybf x dx ,that is ,oOWe say said tothat thediverge.b bxdx=tlim tf xdx，corresp onding improper in tegral con verges. Otherwise,the integral is定义设函数f x在区间上连续，如果反常积分0 :_:.f x dx和 0 f x dx都收敛，则称上述反 常积分 之和为函数f x在无穷区间上的反常积分，记作xdx，即: 0 :_:;f Xdx= J xdx+ 0 f xdx0 t=lim f x dx+ limt .

26、 t t , 0f x dx这时也称反常积分 f x dx收敛；否则就称反常积分OQLet f x be continuous on :匚匕：f bothf x dx发散。0 :f x dx and f x dx con verge, 0bothe n I f x dx is said to con verge and havevalue: 0 :_;-f x dx= J x dx+ 0 f xdx0 t专m t f xdx+tlim .0f xdx.、无界函数的反常积分 (Improper Integra ls of Infinite Integrands)定义 无界函数反常积分 (Impr

27、oper Integra ls of Infinite Integrand)b tf (x)dx = lim f (x)dxa t b-a如果等式右边的极限存在且为有限值 ，此时称反常积分收敛，否则称反常积分发散.Deintion Let f (x) be continuous on the half-open interval la,b and supposethat lim | f (x) .Thent fb tf (x)dx lim f (x)dxa t aProvided that this limit exists and is finite ,in which case we sa

28、y that the integral con verge.Otherwise,we say that the in tegral diverges.无界函数的反常积分 (Improper Integra ls of Infinite Integrands)定义设函数f(x)在半开半闭区间 a,b 1上连续，且 严.“(X)| =：,则b tf (x)dx = lim f (x)dx,a t a如果等式右边的极限存在且为有限值 ，此时称反常积分收敛，否则称反常积分发散.Deintion Let f (x) be continuous on the half-open interval a,b

29、land suppose thatlim(x) =.The ntTb ta f (x)dx=lim f (x)dxa Ja 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.积分函数在内点极限为g的反常积分 (Integrands That are Infinite at an Interior Point)设函数在f (x)在la,b】上除点c(avcb)外连续，

30、且lim f(x)|=，则定义XJcb c ba f (x)dx = a f (x)dx c f (x)dxt b=lim f (x)dx lim f (x) dxtc归 Jc：- t如果等式右边的两个反常积分都收敛，否则称反常积分bf (x)dx 发散.aLet f (x) be con tin uous on la, b .1 except at a nun ber c ,where a ： c . b ,and二：.The n we defi nesuppose thatlim f (x)XTb c bf (x)dx = i f (x)dx 亠 i f (x)dxa a 力t b=lim f (x)dx lim f (x) dxt_C _ a t_c