Closed Formulas :

 
Table 1Constants for Newton-Cotes Closed Formulas

N a wi, i = 0,1,2,3...,N Error
1 1/2 1 1
2 1/3 1 4 1
3 3/8 1 3 3 1
4 2/4 7 32 12 32 7
5 2/288 19 75 50 50 75 19
6 1/140 41 216 27 272 27 216 41
7 7/17280 751 3577 1323 2989 1323 3577 751
8 8/414175 989 5888 -928 10496 -4540 10496 -928 5888 989
9 9/89600 2875 157411080 193445788 5788 193441080 157412857
10 5/299376 16067106300 -48528 272400 -260550 427368 -260550 272400 -48525 106300 16067 



Open Formulas :

Table 2 Constants for Newton-Cotes Open Formulas

N a wi,i = 0,1,2,3...,N Error
1 3/2 0 1 1 0
2 4/3 0 -2 -1 2 0
3 5/24 0 11 1 1 11 0
4 6/20 0 11 -14 26 -14 11 0
5 7/1440 0 611 -453 562 -453 611 0
6 8/945 -594 2196 -2459 2196 -954 460 0


Newton Cotes Open Formular (N = 1)


Example (p. 132) The arc length of a curve in polar coordinate (see fig) is given by


Calculate the arc length of the curve given by  r = 2(1 + cos q); 0 < q < p  by using each of the Newton Cotes Closed Integration Formulas

Solution

กำหนดให้ a = 0, b = p และใช้โปรแกรม 4.2 คำนวณ ผลลัพธ์ที่ได้เป็นดังนี้


ORDER N Integral L

1 8.01823
2 8.00803
3 7.99993
4 7.99996
5 8.00000
6 8.00000
7 8.00000
8 8.00197
9 7.99201
Exact 8.00000