第五章定积分

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()y f x =区间[],a b 上非负、连续，由x a =，x b =，0y =以及曲线()

y 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 interval [],a b . Then the region

bounded by the graph of ()f x , the x -axis, the vertical lines x 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[],a b 上的函数，∆是[],a b 的任意一个分割，

011n n a x x x x b -=<<<<=L ，

其中i x ∆是第i 个小区间的长度，i c 是第i 个小区间的任意一点，那么和

()1

n

i

i

i f c x

=∆∑，1i i i x c x -≤≤

称为黎曼和。

Let ()f x be defined on the closed interval [],a b , and let ∆ be an arbitrary partition

of [],a b ,011n n a x x x x b -=<<<<=L , where i x ∆ is the width of the i th subinterval. If

i c is any point in the i th subinterval, then the sum

()1

n

i

i

i f c x

=∆∑，1i i i x c x -≤≤,

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()f x 在区间[],a b 上有界，在[],a b 中任意插入若干个分点

011n n a x x x x b -=<<<<=L ，把区间[],a b 分成n 个小区间：

[][][]01121,,,,,,,n n x x x x x x -L

各个小区间的长度依次为110x x x ∆=-，221x x x ∆=-，…，1n n n x x x -∆=-。在每个小区

间[]1,i i x x -上任取一点i ξ，作函数()i f ξ与小区间长度i

x ∆的乘积()i i

f

x ξ∆（1,2,,i n =L ），并作出和

()1

n

i i i S f x ξ==∆∑。

记{}12max ,,,n P x x x =∆∆∆L ，如果不论对[],a b 怎样分法，也不论在小区间[]

1,i i x x -上点i ξ怎样取法，只要当0P →时，和S 总趋于确定的极限I ，这时我们称这个极限I 为函数()f x 在区间[],a b 上的定积分（简称积分），记作

()b

a f x dx ⎰，即

()b

a

f x dx ⎰=I =()0

1

lim n

i

i

P i f x ξ→=∆∑,

其中()f x 叫做被积函数，()f x dx 叫做被积表达式，x 叫做积分变量，a 叫做积分下限，

b 叫做积分上限，],a b ⎡⎣叫做积分区间。

Let ()f x be a function that is defined on the closed interval [],a b .Consider a partition

p of the interval [],a b into n subinterval (not necessarily of equal length ) by means of

points

011n n a x x x x b

-=<<<<=L and let

1

i i i x x x -∆=-.On each

subinterval ]1,i i x x -⎡⎣,pick an arbitrary point i ξ(which may be an end point );we call it a sample point for the ith subinterval.We call the sum ()1

n

i

i

i S f x ξ==

∆∑ a Riemann sum for

()f x corresponding to the partition p .

If

()0

1

lim n

i i P i f x ξ→=∆∑exists, we say

()f x is integrable on []

,a b ,where

{}12max ,,,n p x x x =∆∆∆L . Moreover,

()b

a f x dx ⎰,called definite integral (or Riemann

Integral) of ()f x from a to b ,is given by

()b

a

f x dx ⎰=()0

1

lim n

i

i

P i f x ξ→=∆∑.