PHY 480 - Computational Physics
Michigan State University, Spring Semester 2006
INTEGRALS II
Notations:
a = x
°
, b = x
°
+ n h
f
k
= f(x
°
+ k n), f
(p)
(
x
) =
1
n
n
å
i = 1
f
(p)
(
x
i
),
x
i
,
x
Î
(a,b).
Composite Formulae
ó
õ
b
a
f(x) d x = h ( f
1
+ f
2
+ ... + f
n
)
-
h
2
(b
-
a) f
¢
(
x
) + ...
(1)
ó
õ
b
a
f(x) d x = h
æ
ç
è
1
2
f
0
+ f
1
+...+ f
n
-
1
+
1
2
f
n
ö
÷
ø
-
h
2
12
(b
-
a) f
¢¢
(
x
) + ...
(2)
ó
õ
b
a
f(x) d x =
h
3
( f
0
+ 4 f
1
+ 2 f
2
+4 f
3
+ ...+ 4 f
n
-
1
+ f
n
)
-
h
4
90
(b
-
a) f
(iv)
(
x
) + ...
(3)
Formulae with derivatives
ó
õ
b
a
f(x) d x = h
æ
ç
è
1
2
f
0
+ f
1
+...+ f
n
-
1
+
1
2
f
n
ö
÷
ø
+
h
2
12
( f
0
¢-
f
n
¢
) +
h
4
720
(b
-
a) f
(iv)
(
x
) + ...
(4)
ó
õ
b
a
f(x) d x =
h
15
( 7 f
0
+ 16 f
1
+ 14 f
2
+...+ 16 f
n
-
1
+ 7 f
n
) +
h
2
15
( f
0
¢-
f
n
¢
) +
h
6
4725
(b
-
a) f
(vi)
(
x
) + ...
(5)