SIMPSON'S RULE 



249 





Let h be the breadth of a strip, 2n the number of strips, and 

 Vi> y*> !/3 y-tn, y 2n +i the ordinates. 



Considering the first two strips, the ordinates for which are 

 y lt y a , and y 3 . Let OX and OY, the axes of reference, be so 

 chosen that the ordinate y z coincides with the axis OY. There- 

 fore, with reference to hese axes the co-ordinates of the points 



A. H, and C will be ( - h, yj, (0, t/ 2 ), and (h, y 3 ) respectively. 

 Let that part of the curve which passes through the points A, 



B, and C be represented by the equation y = a + bx + ex 2 where 

 a, b, and c are constants. 



These constants can be expressed in terms of the ordinates. 



For when x h, y = y lt and y l = a bh + ch 2 

 when x = 0, y = t/ 2 , and y z = a 

 when x = h, y = y 3 , and y 3 = a + bh + ch 2 



Then 



FIG. 76. 



*- <*-*} 



and c=^{y 1 -2y z + y a } 



Denoting the area of the first two strips by 



(* 



Then A, = I y dx 



J-ft 

 = | (a + bx + ex 2 ) dx 



J h 



ax+ ^bx 2 



m 



= 2ah+- ch 3 

 o 



MF 



o J_, 



