NEWTON-COTE FORMULAS

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