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）设在区间上非负、连续，由，以及曲线所围成的图形称为曲边梯形，其中曲线弧称为曲边。Let be continuous and nonnegative on the closed interval. Then the region bounded by the graph of, the -axis, the vertical

2、lines, and is called the trapezoid with curved edge.黎曼和的定义（Definition of Riemann Sum）设是定义在闭区间上的函数，是的任意一个分割，其中是第个小区间的长度，是第个小区间的任意一点，那么和，称为黎曼和。Let be defined on the closed interval, and let be an arbitrary partition of, where is the width of the th subinterval. If is any point in the th subinterval, t

3、hen the sum ，,Is called a Riemann sum for the partition.二、定积分的定义（Definition of Definite Integral）定义 定积分（Definite Integral）设函数在区间上有界，在中任意插入若干个分点，把区间分成个小区间：各个小区间的长度依次为，。在每个小区间上任取一点，作函数与小区间长度的乘积（），并作出和。记，如果不论对怎样分法，也不论在小区间上点怎样取法，只要当时，和总趋于确定的极限，这时我们称这个极限为函数在区间上的定积分（简称积分），记作，即=, 其中叫做被积函数，叫做被积表达式，叫做积分变量，叫做

4、积分下限，叫做积分上限，叫做积分区间。Let be a function that is defined on the closed interval.Consider a partition of the interval into subinterval (not necessarily of equal length ) by means of pointsand let .On each subinterval,pick an arbitrary point (which may be an end point );we call it a sample point for the i

5、th subinterval.We call the sum a Riemann sum for corresponding to the partition .If exists, we sayis integrable on,where . Moreover,called definite integral (or Riemann Integral) of from to ,is given by =.The equality = means that, corresponding to each >0,there is a such that < for all Rieman

6、n sums for on for which the norm of the associated partition is less than .In the symbol , is called the lower limit of integral , the upper limit of integral,and the integralinterval.定理1 可积性定理 （Integrability Theorem）设在区间上连续，则在上可积。Theorem 1 If a function is continuous on the closed interval ,it is i

7、ntegrable on .定理2 可积性定理（Integrability Theorem）设在区间上有界，且只有有限个间断点，则在区间上可积。Theorem 2 If is bounded on and if it is continuous there except at a finite number of points ,then is integrable on.三定积分的性质（Properties of Definite Integrals）两个特殊的定积分（1）如果在点有意义，则；（2）如果在上可积，则。Two Special Definite Integrals(1) If i

8、s defined at.Then .(2) If is integrable on . Then .定积分的线性性（Linearity of the Definite Integral）设函数和在上都可积，是常数，则和+都可积，并且（1）=;(2) =+; and consequently,(3) =-.Suppose that and are integrable on and is a constant . Then and are integrable ,and (1) =;(2) =+; and consequently,(3) =-.性质3 定积分对于积分区间的可加性（Interv

9、al Additive Property of Definite Integrals）设在区间上可积，且，和都是区间内的点，则不论，和的相对位置如何，都有=+。Property 3 If is integrable on the three closed intervals determined by ，and ,then =+no matter what the order of ，和.性质 4 如果在区间上1，则=。Property 4 If 1 for every in ,then =.性质 5 如果在区间上,则。Property 5 If is integrable and nonne

10、gative on the closed interval ,then .推论1。2 定积分的可比性（Comparison Property for Definite Integrals）如果在区间上，则，。用通俗明了的话说，就是定积分保持不等号。Corollary 1, 2 If and is integrable on the closed interval ,and for all in .Then and 。In informal but descriptive language ,we say that the definite integral preserves inequali

11、ties.性质 6 积分的有界性(Boundedness Property for Definite Integrals )如果在上连续，且对任意的，都有，则。Property 6 If is continuous on and for all in .Then。性质 7 积分中值定理（Mean Value Theorem for Definite Integrals ） 如果函数在闭区间上连续，则在积分区间上至少存在一点，使下式成立=，且=称为函数在区间上的平均值。Property 7 If is continuous on ,there is at least one number bet

12、ween and such that =，and=is called the average value of on .5.2 微积分基本定理（Fundamental Theorem of Calculus）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 (Fundamental Theorem of Calculus) 如果函数在区间上连续,则积分上限函数=在上可导,并且它的导数是=.Theorem 1 Let be continuous on the closed interval and let be a

13、 (variable) point in.Then = is differentiable on ,and=.定理 2 原函数存在定理(The Existence Theorem of Antiderivative)如果函数在区间上连续,则函数=就是在上的一个原函数.Theorem 2 If is continuous on the closed interval ,then = is an antiderivative of on .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 Fundamental Theorem of Ca

14、lculus)如果函数是连续函数在区间上的一个原函数,则 =称上面的公式为牛顿-莱布尼茨公式.Theorem 3 Let be continuous(hence integrable ) on,and let be any antiderivative of on .Then =which is called the Newton-Leibniz Formula. 5.3 定积分的换元法和分部积分法(integration by Substitution and Definite Intgrals by Parts)一. 定积分的换元法(Substitution Rule for Defini

15、te Integrals)二. 定理 定积分的换元法(Substitution Rule for Definite Integrals)假设函数在区间上连续,函数满足条件(1),;(2) 在(或)上具有连续导数,且其值域,则有=,上面的公式叫做定积分的换元公式.Theorem Let have a continuous derivative on (or), and let be continuous on .If , and the range of is a subset of .Then=,which is called the substitution rule for definit

16、e integrals.二.定积分的分部积分法(Definite Integration by Parts)根据不定积分的分部积分法,有 简写为 =或=.According to the indefinite integration by parts ,= = =For simplicity , =or=.5.4 反常积分(Improper Integrals)一.无穷限的反常积分(Improper Integrals with Infinite Limits of integration )定义1 设函数在区间上连续,取,如果极限存在且为有限值,则此极限为函数在无穷区间上的反常积分,记作,即

17、=.这时也称反常积分收敛; 如果上述极限不存在,函数在无穷区间上的反常积分就没有意义,习惯上称为反常积分发散.Let be continuous on ,and .If the limit exists and have finite value , the value is the improper integral of on ,which is denoted by,that is , =,We say that the corresponding improper integral converges.Otherwise ,the integral is siad to diverge

18、. 设函数在区间上连续，取，如果极限存在且为有限值，则此极限为函数在无穷区间上的反常积分，记作，即 =， 这时也称反常积分收敛；如果上述极限不存在，就称反常积分发散。 Let be continuous on，and.If the limit exists and have finite value, the value is the improper integral of on ,which is denoted by ,that is , =，We say that the corresponding improper integral converges. Otherwise, the

19、 integral is said to diverge.定义 设函数在区间上连续，如果反常积分和都收敛，则称上述反常积分之和为函数在无穷区间上的反常积分，记作，即 =+ =+这时也称反常积分收敛；否则就称反常积分发散。Let be continuous on .If both and converge, then is said to converge and have value =+ =+,二、无界函数的反常积分(Improper Integrals of Infinite Integrands)定义 无界函数反常积分(Improper Integrals of Infinite Integrand)设函数 在半开闭区间 上连续,且 ,则 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 be continuous on the half-open interval and suppose that .Then Provided that this limit exists and is finite ,in which case we say that the integral converge.Otherwise,we say th