《定积分的概念和性质Concept.doc
文本预览下载声明
第五章 定积分
Chapter 5 Definite Integrals
5.1 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
显示全部