定积分的概念和性质(Concept.doc

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 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, then 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 ith 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 Riemann 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 integrable 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 is 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（Interval 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 nonnegative 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 inequalities.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 between 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 (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 Calculus)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 Definite 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 definite 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=.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. 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 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 tha