第二节定积分的性质基本内容二、小结思考题经济数学微积分
一、基本内容 二、小结 思考题 第二节 定积分的性质
基本内容("[f(x)± g(x)]dx=(" f(x)dx±/~g(x)dx.性质1eb证[f(x)± g(x)]dx1= limZ[f(5,)± g(5,)]Ax20i=1nnZZ f()Ax;= lim:±limg(5)Ax;1-→02-→0i=1i=1f(x)dx ± ~ g(x)dx.（此性质可以推广到有限多个函数代数和的情况）经济数学微积分
证 [ ( ) ( )]d b a f x g x x i i i n i = f g x = → lim [ ( ) ( )] 1 0 i i n i = f x = → lim ( ) 1 0 i i n i g x = → lim ( ) 1 0 ( )d b a = f x x ( )d . b a g x x [ ( ) ( )]d b a f x g x x ( )d b a = f x x ( )d b a g x x . （此性质可以推广到有限多个函数代数和的情况） 性质1 一、基本内容
性质2kf(x)dx = k (f(x)dx(k为常数)nZkf(5)Ax;kf(x)dx = lim证20i1nnZf(5)Ax;Z f(5)Ax; = klim= limk1→020i=1i1f(x)dx.微积分经济数学
( )d ( )d b b a a kf x x k f x x = ( k为常数). 证 ( )d b a kf x x i i n i = kf x = → lim ( ) 1 0 i i n i = k f x = → lim ( ) 1 0 i i n i = k f x = → lim ( ) 1 0 ( )d . b a = k f x x 性质2
性质3假设a<c<b[" f(x)dx= f° f(x)dx + f' f(x)dx.补充：不论a,b,c的相对位置如何，上式总成立例若 a<b<cJ° f(x)dx = " f(x)dx + f' f(x)dx则 f~ f(x)dx= J° f(x)dx-J’ f(x)dxf f(x)dx+ f" f(x)dx.（定积分对于积分区间具有可加性）微积分经济数学
( )d b a f x x ( )d ( )d c b a c = + f x x f x x . 补充：不论 a,b,c 的相对位置如何, 上式总成立. 例 若 a b c, ( )d c a f x x ( )d ( )d b c a b = + f x x f x x ( )d b a f x x ( )d ( )d c c a b = − f x x f x x ( )d ( )d . c b a c = + f x x f x x （定积分对于积分区间具有可加性） 则 性质3 假设a c b
性质4"1.dx=dx=b-a.0性质5如果在区间[a,bl上f(x)≥0,则" f(x)dx ≥0. (a<b)证 f(x)≥0, f(,)≥0, (i=1,2,..,n):: Zf(5,)Ax, ≥0,: △x, ≥ 0,i=1a = max[Ax,Ax2,".",Ax.]nZ f(5,)Ax; = (" f(x)dx ≥ 0.lim2-0i1化经济数学微积分
1 d b a x d b a = x = b − a. 则 ( )d 0 b a f x x . (a b) 证 f (x) 0, ( ) 0, i f (i = 1,2, ,n) 0, xi ( ) 0, 1 = i i n i f x max{ , , , } = x1 x2 xn i i n i f x = → lim ( ) 1 0 ( )d 0. b a = f x x 性质4 性质5 如果在区间[a,b]上 f (x) 0