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Integration Newton-Cotes formulas

A set of very popular integration formulas are known as classical Newton-Cotes formulas for equally spaced step. The general formula is:

where bi are specific coefficient for any degree n. Several methods exists to find the coefficients:

· Indefinite coefficient system
· Lagrange interpolation function

Here give only same outline of the first method.It requires to solve the following system to find all coefficients for e given degree n

Example: for finding the 2° degree Newton-Cotes formula (Cavalieri-Simpson)

Into the table below we can find all coefficients for formulas from 1° to 10 degree. For clarity, coefficients are made integer by mean of an appropriate constan K

Degree =>

1

2

3

4

5

6

7

8

9

10

K

2

3

8

45

288

140

17280

14175

89600

299376

b0

1

1

3

14

95

41

5257

3956

25713

80335

b1

1

4

9

64

375

216

25039

23552

141669

531500

b2

1

9

24

250

27

9261

-3712

9720

-242625

b3

3

64

250

272

20923

41984

174096

1362000

b4

14

375

27

20923

-18160

52002

-1302750

b5

95

216

9261

41984

52002

2136840

b6

41

25039

-3712

174096

-1302750

b7

5257

23552

9720

1362000

b8

3956

141669

-242625

b9

25713

531500

b10

80335

 

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