定积分概念与性质(Concept.doc
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1、定积分概念与性质(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
2、, 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
3、 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
4、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
5、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
6、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 ,i
7、t 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
8、) 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(In
9、terval 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
10、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 in
11、equalities.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 numb
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