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()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 regionbounded 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()1niii 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 partitionof [],a b ,011n n a x x x x b -=<<<<=L , where i x ∆ is the width of the i th subinterval. Ifi c is any point in the i th subinterval, then the sum()1niii 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ix ∆的乘积()i ifx ξ∆（1,2,,i n =L ），并作出和()1ni 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()ba f x dx ⎰，即()baf x dx ⎰=I =()01lim niiP 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 partitionp of the interval [],a b into n subinterval (not necessarily of equal length ) by means ofpoints011n n a x x x x b-=<<<<=L and let1i i i x x x -∆=-.On eachsubinterval ]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 ()1niii S f x ξ==∆∑ a Riemann sum for()f x corresponding to the partition p .If()01lim ni 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,()ba f x dx ⎰,called definite integral (or RiemannIntegral) of ()f x from a to b ,is given by()baf x dx ⎰=()01lim niiP i f x ξ→=∆∑.The equality ()01limniiP i f x ξ→=∆∑=L means that, corresponding to each ε>0,there is a0δ> such that()1niii f x L ξ=∆-∑<ε for all Riemann sums ()1niii f x ξ=∆∑ for ()f xon [],a b for which the norm P of the associated partition is less thanδ.In the symbol()baf x dx ⎰, a is called the lower limit of integral , b the upper limitof integral ,and [],a b the integralinterval.定理1 可积性定理 （Integrability Theorem ）设()f x 在区间[],a b 上连续，则()f x 在[],a b 上可积。

Theorem 1 If a function ()f x is continuous on the closed interval [],a b ,it is integrable on [],a b .定理2 可积性定理（Integrability Theorem ）设()f x 在区间[],a b 上有界，且只有有限个间断点，则()f x 在区间[],a b 上可积。

Theorem 2 If ()f x is bounded on [],a b and if it is continuous there except at a finite number of points ,then ()f x is integrable on [],a b .三．定积分的性质（Properties of Definite Integrals ） 两个特殊的定积分（1）如果()f x 在x a =点有意义，则()0aaf x dx =⎰；（2）如果()f x 在[],a b 上可积，则()abf x dx =⎰-()baf x dx ⎰。

Two Special Definite Integrals (1) If ()f x is defined at x a =.Then()0aaf x dx =⎰.(2) If ()f x is integrable on [],a b . Then()a bf x dx =⎰-()baf x dx ⎰.定积分的线性性（Linearity of the Definite Integral ）设函数()f x 和()g x 在[],a b 上都可积，k 是常数，则()kf x 和()f x +()g x 都可积，并且（1）()bakf x dx ⎰=()bak f x dx ⎰;(2) ()()baf xg x dx +⎡⎤⎣⎦⎰=()b a f x dx ⎰+()ba g x dx ⎰; and consequently,(3)()()b af xg x dx -⎡⎤⎣⎦⎰=()ba f x dx ⎰-()ba g x dx ⎰.Suppose that ()f x and ()g x are integrable on [],a b and k is a constant . Then()kf x and ()()f x g x + are integrable ,and(1)()bakf x dx ⎰=()bak f x dx ⎰;(2) ()()ba f x g x dx +⎡⎤⎣⎦⎰=()ba f x dx ⎰+()ba g x dx ⎰; and consequently, (3)()()ba f x g x dx -⎡⎤⎣⎦⎰=()ba f x dx ⎰-()ba g x dx ⎰. 性质 3 定积分对于积分区间的可加性（Interval Additive Property of DefiniteIntegrals ）设()f x 在区间上可积，且a ，b 和c 都是区间内的点，则不论a ，b 和c 的相对位置如何，都有()caf x dx ⎰=()b af x dx ⎰+()cbf x dx ⎰。

Property 3 If ()f x is integrable on the three closed intervals determined by a ，b ，andc ,then()caf x dx ⎰=()b af x dx ⎰+()cbf x dx ⎰no matter what the order of a ，b ，和c .性质 4 如果在区间[],a b 上()f x ≡1，则1badx ⎰=badx ⎰=b a -。

Property 4 If ()f x ≡1 for everyx in [],a b ,then1badx ⎰=badx ⎰=b a -.性质 5 如果在区间[],a b 上()f x ≥0,则()baf x dx ⎰≥0()a b <。