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新编文档-第一节定积分的概念与性质(1)-精品文档_图文

新编文档-第一节定积分的概念与性质(1)-精品文档_图文







1

y
B

o

A
a

C D
C E x
b

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yf(x)

x x ax b



A?

.

oa

bx





y

y

oa

b xo a

bx













[a,b ] a x0 x1x2 xn1xnb,
[a,b]n y
[xi1, xi ] xi xi xi1;

[ xi1, xi]
i

oa

x1

b xi1 i x i xn1

x

[xi1,xi] f(i )

A if(i) xi







n

Af(i )xi



i1

,
max{x1,x2,xn} ( 0)

n



Alim
0i1

f(i)xi





2
v v(t) [T1 ,T2 ] t v(t) 0.



(1) (2)

T 1 t 0 t 1 t 2 t n 1 t n T 2 ti titi1
siv(i) ti



(3) (4)

n
s v(i )ti
i1
m t 1 ,a t2 , x , tn } {

n
slim 0i1v(i)ti





f ( x)[a,b][a,b]

a x x x x x b

012

n 1 n

[ a ,b ] n

xi xi xi1(i 1,2,)
i i xi f ( i ) x i( i 1 , 2 , )
n
Sf(i)xi
i1
max{ x1, x2 ,, xn } [a,b]



[ x i 1 ,x i]
i 0 S
I I f ( x)

[a,b]



b

n

af(x )d x I l i0i m 1f(i) x i




















[a,b]





(1)

.

b

b

b

a f (x)dxa f(t)dt a f(u)du

(2)i .

I



(3)0n n0
(4)f(x)[a ,b]

f (x) [a ,b].
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1 f ( x)[a, b]

f( x ) [ a ,b ] .

2 f(x ) [ a ,b ]



f ( x)



[ a ,b ] .

f (x)[a ,b]

f (x)[a ,b]





f(x)0,

b
a f(x)dxA



f(x)0,

b
a f(x)dxA



A1 A2

A3 A4

b
af(x )d x A 1 A 2A 3 A 4



x f(x)
xa, xb x x











1

1
0

[0,1] n

x 2dx
xi

.
i
n

(

i



1,2,, n)

[ xi1,

xi ]xi



1 (i n



1,2,, n)

i x i ( i 1 ,2 , ,n )

n

n

n

f (i )xi i2xi xi2xi ,

i 1

i1

i 1



n



i

2




1

i1 n n



1 n3

n
i2
i1

n 13n(n1)62 (n1)

161n12n1, 0 n

1 x2dx 0

n



lim
0 i1

i2xi

lim 11121 n6 n n

1. 3



2. :

1 n

i

(1) lim 1

n n i1

n

1p2pnp

(2) lim n

np1

:

1n (1) lim

1 i

n
lim

1 i 1

n n i1

n

n i1

nn

x i

1
0 1xdx
O

i

i1 i

1x

nn

1p2pnp

(2) lim n

np1

n
lim n i1

i n

p1 n

x i

1x p dx 0

i




f(x)C[a,b],abf(x)dx ,

:

y

[a , b] n : x ba ,
n
x i a i x (i 0 ,1 , ,n )

f(x i) y i (i 0 ,1 , ,n ) 1.

O

a

xi1x i

bx

b
a

f

( x) dx

y 0 x y 1 x y n 1 x

2.

b n a(y 0y 1 y n 1 )

b
a

f

(

x)

dx

y 1 x y 2 x y n x

b n a(y 1y2 yn )



3.

b

y

a f (x)dx



n i1

1

[ 2

yi1



yi

]x

O a x i 1x i

bx

b n a1 2(y0yn ) (y 1 yn 1)

y

4.

b

f (x)d x
a
ba

m

m 1

O

a x0

x2i2x2i1x 2 i

6 my0y2m 4i 1y2i 12i 1y2i

bx x2m



3. i xi



I

14 01x2

dx.

0 1

0.0 0.1

( n = 10, 5)

2 0.2

:yi()

3 0.3 4 0.4



I3.139935 0.5

6 0.6

I3.141597 0.7



8 0.8

I0 11 4x2dx3.1415926

9 0.9 10 1.0

yi 4.00000 3.96040 3.84615 3.66972 3.44828 3.20000 2.94118 2.68456 2.43902 2.20994 2.00000







1a



b b a

f

(

x)dx



0

2 a b a b f(x ) d x b af(x ) d. x





1ab[

f

(

x)



g(

x)]dx



b
a

f

(

x)dx



b
a

g(

x)dx

ab[f(x)g(x)d] xl i0m i n1[f(i)g(i) ]xi

n

n

lim 0 i1

f(i

)xi

lim 0 i1

g(i

)xi

b a

f

b
(x)dxa g(x)dx.





2

b
kf ( x)dx k

b
f ( x)dx

( k ).

a

a



abkf(x)dxlim 0in1kf(i)xi

n
limk
0 i1

f(i)xi

n
klim
0i1

f(i)xi

k

b
f(x)dx.
a

3

a c ba bf(x )d x a cf(x )d x c bf(x )d.x

a,b,c, .

abc,

c

b

c

a f (x)dx af(x)d xbf(x)dx



b
a

f

(x)dxa cf(x)d xb cf(x)d xa cf(x)d xcbf(x)d.x

b f(x ) d x cde b

a

ac d

l



4 a b1d x a bd x ba.
5----

[ a ,b ] f(x ) 0 ab f ( x)dx 0. (a b)

f(x)0,f(i)0, (i1 ,2, ,n )

n
xi 0, f(i)xi 0, m x 1 , a x 2 , x , x n } {

i1

n
lim
0 i1

f(i

)xi

b
f(x)dx0. a

--

1 2 e xdx 2 xdx.

0

0

f(x)exx, x[2,0]

f(x)0, 0(exx)dx 0, 2



0 exdx 2

0
xdx,
2



2exdx

2
xdx.

0

0



5 1 [ a , b ] f ( x ) g ( x )



b
a

f

(

x )dx

b
a g(

x)dx.

(ab)

f(x )g (x ), g (x ) f(x ) 0 ,

b

b

b

a[g(x)f(x)d ] x0, ag (x)d x af(x)d x 0 ,

a b f ( x ) d x a b g ( x ) d . x

2 a bf(x)d xa bf(x)d.x(ab)

f( x ) f( x ) f( x ) ,

a b f(x ) d x a b f(x ) d x a b f(x ) d , x a bf(x)d xa bf(x)d.x
f(x)[a,b]f( x)

[a,b]. .



6

M m f ( x)[a, b]



m(b



a)



b
a

f

(

x)dx



M (b



a).

m f(x )M , a bm d xa bf(x)d xa bM,dx

b
m (b a ) af(x ) d x M (b a ).



2

0

3



1 sin3

dx . x

.

f(x)3s1in3x, x [0,], 0si3nx1,

143s1in 3x13,

0 1 4 d x0 3s1i3x n d x0 1 3 d,x

40 3s1i3n xd x3.





3

2

f(x)sinx,4 x

sin xdx. x
x [, ] 42

f(x)xcoxxs 2six ncoxs(xx2taxn) 0,

f ( x)[ , ], x x ,

42

4

2

Mf()2 2, 4

m f(2)2,

ba,
24 4

244 2sx ix n dx 224,

1 2


2

4

sinxdx x

2. 2



7 f(x ) [ a ,b ]

[a, b]



b
a

f

(

x)dx



f ( )(b a).

(a

b)



m ( b a ) a b f(x ) d M x ( b a )

1b



mbaaf(x)d xM



[a, b]

f()b 1aabf(x)d,x



b
a

f

(

x

)dx



f ( )(b a).

(ab)





y

[a, b]

f ()

[a, b] yf(x)



f()
o a b x
1. [a,b], (a ,) b
(a,b)

2.a>ba<b.

3.

f() 1

b
f(x)dx

baa

f (x) [a,b].

f (x) [a,b] .





4 f ( x) lim f ( x) 1 x

lim

x2
t sin

3

f

(t )dt .

x x

t



[x,x2],

xx2tsin3t f(t)dtsi 3 nf()(x2x), xl im xx2tsin 3t f(t)dt 2l imsin3f()

2lim 3f()6.















----





1.n l in 1 m si n n si2 n n si(n n n 1 ) .

n l in 1 m s n i n s2 i n n s( in n n 1 ) sn i n n



lim1 n n n i1

sini n

1nl i min1sininn

1


sinxdx.

0

i xi





n l in 1 m s n i n s2 i n n s( in n n 1 ) sn i n n



lim1 n n n i1

sini n

n
lim

sini 1

ni1 n n

1sin(x)dx. 0

i xi




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