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Simpson's Rules for areas and centroids 69
The area of the elementary strip is ydx. Then the area enclosed by the curve and the axes of reference is given by
2h
ydx
a0 a1x a2x2
a0x 2 3 O
2a0h 2a1h2 83 a2h3
Area of figure But y ; Area of figure

Assume that the area of figure
O
2h O
a0 a1x a2x2 dx a1x2 a2x3 2h
Ay1 By2 Cy3
Using the equation of the curve and substituting `x' for O, h and 2h
respectively:
Area of figure Aa0 B a0 a1h a2h2 C a0 2a1h 4a2h2
a0 A B C a1h B 2C a2h2 B 4C
;2a0h 2a1h2 83a2h3 a0 A B C a1h B 2C a2h2 B 4C
Equating coef®cients:
A B C 2h, B 2C 2h, and B 4C 83 h
From which:
A h; B 4h; andC h 333
; Area of figure h3 y1 4y2 y3 This is Simpson's First Rule.
It should be noted that Simpson's First Rule can also be used to ®nd the area under a curve of the third order, i.e., a curve whose equation, referred to the co-ordinate axes, is of the form y a0 a1x a2x2 a3x3, where a0, a1, a2 and a3 are constants.