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MATLAB Simpson's Rule

The trapezoidal and Simpson's rules are special cases of the Newton-Cote rules which use higher degree functions for numerical integration.

Let the parabola represent the curve of the figure.

y=αx2+βx+γ…………..equation 1

MATLAB Simpson's Rule

The area under this method for the interval -h≤x≤h is

MATLAB Simpson's Rule

The curve passes through the three points(-h,y0 ),(0,y1 ),and (h,y2 ). Then, by an equation, we have:

MATLAB Simpson's Rule

We can now evaluate the coefficients α,β,γ and express equation2 in terms of h, y0, y1and y2.

This is done with the following procedure.

By substitution of (b) of equation3 into (a) and (c) and rearranging we obtain

αx2-βh=y0-y1…..equation4

αx2+βh=y2-y1…..equation5

Adding of equation4 with equation 5 yields

2αh2=y0-2y1+y2…...equation6

and by substitution into equation2, we obtain

MATLAB Simpson's Rule

Now, we can apply equation8 to successive segments of any curve y=f(x) in the interval a≤x≤b as shown on the curve of the figure.

MATLAB Simpson's Rule

We observe that a parabola can approximate each segment of width 2h of the curve through its ends and its midpoint. Thus, the area under segment AB is

MATLAB Simpson's Rule

Likewise, the area under segment BC is

MATLAB Simpson's Rule

and so on. When the areas under each segment are added, we obtain

MATLAB Simpson's Rule

Since each segment has width 2h, to apply Simpson's rule of numerical integration, the number n of subdivisions must be even. This restriction does not apply to the trapezoidal rule of numerical integration.

The value of for equation11 is found from

MATLAB Simpson's Rule
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