定积分的概念和性质(Concept

1、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 intervala,bL Then the region bo

2、unded bythe graph of f x , the x -axis, the vertical lines x 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，xna =冷 ： Xi ： | | : 人其中 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 Ac 兰洛i V称为黎曼和。Let f x

3、be 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 sumj f ( c Hxi ， xi i TIs called a Riema nn sum for the partiti on二、定积分的定义 ( Definition

4、of Definite Integral )定义 定积分 ( Definite Integral )设函数fx在区间！a,b 丨上有界，在 a,b中任意插入假设干个分点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 ,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，如果不管对a,b怎样分法，也不管在小区间txAxj上点？怎样取法，只要当 P t0时，和S总趋于确定的极限I,这时我们称

5、这个极限I为 函数f x在区na fxdx=l = IPmA f i间l.a,b 1上的定积分简称积分，记作:f x dx，即x叫做积分变量，a叫做积分下限，其中f x叫做被积函数，f x dx叫做被积表达式，b叫做积分上限，| a, b 叫做积分区间。Let f x be a fun cti on that is defi ned on the closed in tervalla, b .C on sider a partiti onpointS =xo : 为：川：:XnXn = bandlet 为=为一x.Oneachsub in terval 內， Xj J 1 ,pick an a

6、rbitrary pointi (whichmay be an end point );wecall it ans 八 f i Xi a Riemannidb,b J into n sub in terval (not n ecessarily of equal len gth ) by means ofsample point for the ith sub in terval.We callp of the in tervalIfnf(勺岁 x exists,wethe sumsum forsay f x isin tegrable on a,bl,wherebf x dx ,called

7、 definite integral (or Riemannap =max? M, X2,lll, Xn< . Moreover,f xcorresp onding to the partiti onIn tegral) off x from a to b ,is give n byfxdx=ipmo faThe equalityHPi . Xi = L means that, corresponding to each； >0,there is an nd >0 such that 瓦 f ( £yiXj -L for all Riemann sums 瓦 f (

8、? yiXjfor f (x)i =1i =1on a, b for which the norm P of the associated partition 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，贝U f x在la,b 上可积。Theore

9、m 1 If a function f x is continuous on the closed interval I a, b 1 ,it is integrable on I a, bI.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，贝f x 在区间 la,b 上可积。Theorem 2 If f x is bounded on la, b 丨 and if at a finite it is con ti nu ous there except number of points ,then f x is

10、integrable on a,b .L三. 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=a点有意义，那么.f x dx = 0 ；a(2)如果 f x 在 l.a,b 1 上可积，贝 f x dx 二- f x dxb' aTwo Special Definite Integralsa(1) If f x is defined at x=a.Then . f x dx = 0.af x dx = f x dx . b' a)(2) If f x is integrable on !a,bL Th

11、en 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，那么kf x和f x + g x都可积,并且bb(1) kf x dx= k f x dx;aab b| f x 亠 g x dx= f x dx+bg x dx; and consequently,a_f x-g x dx =它 f xdx ag X dX.b b bSuppose that f x and kf x and f x g xare in tegrable ,andg x are integrable on a,b l and k i

12、s a constant . Thenb b(1) a kf x dx= ka fx dx;b b bII f XAg 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 DefiniteIn tegrals )设f x在区间上可积，且 a , b和c都是区间内的点，那么不管 a , b和c的相对位置cbc如何，都有 . f xdx=. f x dx+ . f x dx。a，a' a&

13、#39; bvProperty 3 If f x is integrable on the three closed intervals determined by b ， and c ,thencbca 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= dx = b-a。“= aProperty 4 If f x _1 for every x in a, bl,thenbb1dx= dx = b - a .a' a性质 5如果在区间

14、 l.a, b 1 上 f x _ 0,那么 f x dx 丄0 a b 。Property 5 If f x isintegrable 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 DefiniteIntegrals )如果在区间l.a,b 1上，f x岂g x，贝Ua f xdx E ag xdx ，L f(x)dx 兰|f(xOx o用通俗明了的话说，就是定积分保持不等号。Corollary 1, 2 If f x

15、 andg X)is integrable on the closed intervalfa,b】 , andf x _g x for all x in l.a,b l.Then b a f xdx <bag xdxandf (x)dx 兰 jjf (x jdx 。In in formal but descriptive Ian guage ,we say that the in equalities.defi nite in tegralpreserves性质 6 积分的有界性 (Boundedness Property for Definite Integrals )如果 f x 在

16、 la,b 上连续，且对任意的l.a,b 1 ，都有 m 乞 f x < M ，那么bm b - a I f x dx 辽 MProperty 6 If f x is continuous on l.a,b 1and m _ f x - M for all x inl.a, b l.Then性质 7m b - a j I f x dx 乞 M b - ao积分中值定理 (Mean Value Theorem for DefiniteIntegrals )如果函数 f x 在闭区间 la,b 1 上连续，那么在积分区间 a,b 1 上至少存在一点 ?，使下式成立ba f xdx= f b-

17、a，1 b f =L a f xdx b _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 numberbetweena and b such thatL f (x)dx = f 仁)(b-a),aanda f XdXis called the average value off x on a,b.5.2 微积分根本定理(Fun dame ntal Theorem of Calculus).积分上限的函数及其导数 (Accumulation Function and I

18、ts Derivative)定理 1 微积分根本定理(Fu ndame ntal Theorem of Calculus)如果函数f X在区间la,b吐连续,那么积分上限函数且它的导数是XX f t dt在la,b 上可导，并'avXd f f (t dtx =f x a 乞 x 乞 b .Theorem 1 Let f x be con ti nu ous on the closed in tervala,bl and let x be a (variable)poi ntin a, b .The nX! f t dt is? avdiffere ntiableonx=f x a E

19、x Ebd J f (t dtla,b l,and adx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，贝U函数'x = Xf t dt就是f x在l.a,b 11a上的一个原函数？vTheorem 2 If f x is continuous on the closed interval1xa,bthenX 1.aisan antiderivative of f x on la, b I.二.牛顿-莱布尼茨公式(Newton丄eibniz Formula)定理 3 微积分第一根本定

20、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,那么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 ) on a,bl ,and let F X be any antiderivative off X on La, b) .Thena f xdx=F b -Fwhich is called the Newton-Leibniz Formula5.3 定积分的换元法和分部积分法 (in

21、tegration by Substitution and Definite Intgrals by Parts)一 . 定积分的换元法 (Substitution Rule for Definite Integrals)二 .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,那么有b:a fXdX= f t' t dt, 上面的公式叫做定积分的换元公式Theo

22、rem 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.Thenba f xdx= ：:f t ' t dt which is called the substitution rule for definite二.定积分的分部积分法integrals.(Definite Integration by Parts)u x v' x dx =bu x v

23、' x dx a根据不定积分的分部积分法 ， 有二 u x v x u'xvxdx-| u x v xb-f v(x )u'( x Jdx简写为b? b buv'dx =uv ，vu'dxaaaudv= I uv I - vdu a aAccording to the indefinite integration by parts ,fu( xy( x 加ju x)vY x)dx=u( x)v( x)- Ju'( x)v( x)dxbb- av x u' xdx=u x)v( x)For simplicity ,uv'dx= I

24、uv 1 ab-vu'dx aorb budv= I uv -vdu.5.4 反常积分 (Improper Integrals).无穷限的反常积分 (Improper Integrals with Infinite Limits of integration )定义 1 设函数f x在区间a,-" ：?上连续，取t a ,如果极限tim_ f x dx 存在且为有限值 ，那么此极限为函数fx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Laf x dx 收敛；如果上述极限不存在，函数 f x 在

25、无穷区间a,上的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 continuous on |a, ： ,and t a .If the limit pm 一. f x dxhave finite value , the value is the improper integral of f x on 2, 亠i ,which-beby f x dx ,that is ,a:tf x dx= lim f x dx,at ： . aWe say that the corresponding improper integral converg

26、es Otherwise ,the integraldiverge .exists andis deno tedis siad to设函数f (x j在区间(-°b 上连续，取tcb，如果极限lim f(xpx存在且为有限值那么此极限为函数fx在无穷区间-:，b 1上的反常积分，记作b fx dx，即bf xdx =tlimbf xdx ， t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OQf x dx 发散。Let f x be continuous onbandt : b.If the limit lim t f x dxexists and havef

27、inite value, the value isthe improper integral of f x on 一匚亠 b|,whichis denoted byb,that is ,f x dxoObxdx =tlim tf xdx ，We say said tothat thediverge.定义收敛，那么corresp onding improper设函数 f x 在区间上连续，如果反常积分in tegral con verges. Otherwise,0 :_:.f x dx 和the integral is0 f x dx 都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

28、，记作xdx，即: 0 :_:;f Xdx= J xdx+ 0 f xdx0f x dx=lim f x dx + limt . tt ,这时也称反常积分f x dx收敛；否那么就称反常积分OQLet f x be continuous on : 匚匕: f bothbothe n I f x dx is said to con verge and havef x dx 发散。0 :f x dx and f x dx con verge,0value: 0 :;-f x dx= J x dx+ 0 f xdx0t专 m t f xdx +tli m .0fxdx.、无界函数的反常积分 (Imp

29、roperIntegra ls of Infinite Integrands)定义 无界函数反常积分 ( Improper Integra ls of Infinite Integrand)f (x)dx = lim f (x)dxt 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,否那么称反常积分发散DeintionLet f (x) be continuous on the half-open intervalla,b and supposethat lim | f (x) .Thent,f (x)dx lim f(x)dProvided that this limit exists and is finite ,in which case we say that the integral verge.Otherwise,we say that the in tegral diverges.con无界函数的反常积分 (Improper In tegrals of Infin ite In tegra nds)定义 设函数f(x)在半开半闭区间a,b 1上