.
1
8
.
1
(
) 1
) ( )
( )
( x dx F b F a
b f
a = -
ò
ò
= b
a f x dx
I ( )
. ( .
F
.
2
: f(x)
ò
ò a b
b
a dx
x dx x
x sin
, sin 2
f(x)
x i x 0 x 1 … x n
y i = f(x i ) y 0 y 1 … y n
.
3
? .
x y
: 1
1 [ ( ) ( )]
) 2
( b a f b f a
dx x
b f
a - +
ò »
: 2 ) 3
a b
.
4
2,(
i i n
i
i i
b
a f x dx = å f D x = D x
ò ( ) lim ® = ( ) , max
0 1 x l
l
2, å ) = ( 0
= n D
i
i
i x
f
1
) ( x
= 2,(
2 = 2, ) x k ( = ( 0 2,(
ò å
=
» n
k
k k
b
a f x dx A f x
0
) (
)
(
5
L :
1. ?
2. ?
3. .
4. .
0 5
1.5
ò
= b
a f x dx
S ( )
.
6
h x
f S
n j
j
n å
=
=
1
) (
ò å
® =
= n
j h j
b
a f x dx f x h
0 1 ( )
lim )
(
h 1 0.5 0.2 ···
S n 5.2908 5.1044 4.9835 ···
0 1 2 3 4 5 6 7 8 9 10
0 0.5 1 1.5
) 1 (
3
= x - e x x
f
.
7
] [ )
( )
(
0
f R x
f A dx
x f
n
k
k k
b
a = å +
ò =
R[f ] ——
x 0 , x 1 , ···, x n ——
A 0 , A 1 , ···, A n ——
: a b
)]
( )
( 2 [
)
( b a f a f b
dx x
b f
a - +
ò » A 0 = (b – a )/2
A 1 =(b – a )/2
) ( )
( )
( x dx A 0 f a A 1 f b
b f
a » +
ò
8
.
[a b] : a ≤ x 0 < x 1 < x 2 < …… < x n ≤ b
å =
» n
j
j
j x f x
l x
f
0
) (
) ( )
(
å ò
ò =
» n
j
j b
a j b
a f x dx l x dx f x
0
) (
] )
( [
) (
) , ,
2 , 1 , 0 (
, )
( x dx j n
l
A b
a j
j = ò = !
] [
) (
) (
0
f R x
f A dx
x f
n
j
j j
b
a = å +
ò =
Lagrange
.
9
ò
ò - = + + +
= b
a n
b n
a n x dx
n dx f
x L
x f f
R ( )
)!
(
) )] (
( )
( [ ]
[
) (
1 1
1 x w
2.
) 2 (
1
0 dx b a
a b
x A b b
a = -
-
= ò - A 1 = ò a b b x - - a a dx = 2 1 ( b - a )
] [ )]
( )
( 2 [
)
( b a f a f b R f
dx x
b f
a - + +
ò =
è
.
10
3. .
è x 0 = a
2 / ) ( b a h = - , x k = x 0 + kh ( k = 0, 1, 2 )
ï ï ï î ï ï ï í ì
- -
=
- -
-
=
- -
=
) )(
2 ( ) 1
(
) )(
1 ( )
(
) )(
2 ( ) 1
(
1 2 0
2
2 2 0
1
2 2 1
0
x x
x h x
x l
x x
x h x
x l
x x
x h x
x l
) )(
)(
! ( 3
) ) (
( ) ( )
( 0 1 2
2 0
x x
x x
x f x
x f x l
x f
k
k
k ¢¢¢ - - -
+
= å
=
x
ï ï î ï ï í ì
=
=
=
ò ò ò
b a
b a
b a
dx x
l A
dx x
l A
dx x
l A
) (
) (
) (
2 2
1 1
0 0
] [ )
( )
( )
( )
( x dx A 0 f x 0 A 1 f x 1 A 2 f x 2 R f
b f
a = + + +
ò
.
11
h dx
x x
x A x x
ò x - -
= 2
0 2
2 1
0 2
) )(
(
h dx
x x
x A x x
ò x - - -
= 2
0 2
2 0
1
) )(
(
h dx
x x
x A x x
ò x - -
= 2
0 2
1 0
2 2
) )(
(
] [ )]
( 2 )
( 4 )
( 6 [
)
( a b f b R f
f a
a f dx b
x
b f
a + + +
- +
ò =
Simpson ,
ò - - -
= b
a f x x x x x x dx
f
R ( )( )( )( )
! 3 ] 1
[ ( 3 ) x 0 1 2
) 6 (
1 b - a
=
) 6 (
1 b - a
=
) 3 (
2 b - a
=
12
4. .
a b
h = -
) ( )
( )
( )
( )
( )
( )
( )
( )
( x f a 0 x f b 1 x f a 0 x f b 1 x
H = a + a + ¢ b + ¢ b
:
ò
ò = + -
= 1
0
2 0
0 a ( x ) dx h ( 1 2 x )( 1 x ) d x
A b
a
ò
ò = + -
= 1
0
2 1
1 a ( x ) dx h [ 1 2 ( 1 x )] x d x
A b
a
2
= h
h a
x ) /
( -
x =
2
= h ]
[ )
( )
( x dx H x dx R f
f b
a b
a = ò +
ò
) ( )
( )
( )
( )
( x dx A 0 f a A 1 f b B 0 f a B 1 f b
b H
a = + + ¢ + ¢
ò
.
13
] [ )]
( )
( 12 [
)]
( )
( 2 [
) (
2
f R a
f b
h f b
f a
h f dx
x
b f
a = + - ¢ - ¢ +
ò
ò - -
= b
a f x a x b dx
f
R ( 4 ) ( )( ) 2 ( ) 2
! 4 ] 1
[ x
ò
ò = -
= 1
0
2 2
0
0 b ( x ) dx h x ( 1 x ) d x
B b
a
x x
x
b x dx h d
B b
a ò
ò = - -
= 1
0
2 2
1
1 ( ) ( 1 )
12
= h
12 - h
=
: m f(x), .
. b f ( x ) dx b a [ f ( a ) f ( b )]
a - +
ò » 2 1
ò å
=
» n
k
k k
b
a f x dx A f x
0
) (
) (
m
m+1 , L
: Simpson 3 What? Why?
R[f ] =
Z b
a
f (x)dx
X n
k=0
A k f (x k ) ⌘ 0
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