What is Simpson's Rule and how can we derive it?

Thanks

simpson's rule is almost the trapezium rule, except that it is based on the area under parabolas instead of trapezia

For a parabola on the interval [-h,h]:

ò_-h^h (a x²+b x+c) dx=(3/h)(y₀ + 4y₁ + y₂)

where if f(x) = a x²+b x+c

f(-h)=y₀, f(0)=y₁, f(h)=y₂
this can be proven, if necessary, by inspection of the general case.

Therefore, if we divide a curve into AN EVEN number of sub-intervals of equal length, its area can be approximated (with more accuracy in most cases than the trapezum rule), with the sum of the areas under parabolas extending across PAIRS of subintervals using the following rule.

ò_a^b g(x) dx=h/3 ((y₀+4y₁+y₂)+(y₂+4y₃+y₄)+(y₄+4y₅+y₆)+¼+ (y_n-3+4y_n-2+y_n-1)+(y_n-2+4y_n-1+y_n))

But as every y_i where i is even occurs twice, this can be reduced to the following:

ò_a^b g(x) dx=h/3(y₀+4y₁+2y₂+4y₃+2y₄+¼+2y_n-2+4y_n-1+y_n)

where y_i is the value of g(x) at the partition point x_i=a+i×h

for the approximation to work:

n is even, and h=(b-a)/n

Thanks a lot...
Yeah, it seems to be a lot more accurate than the trapezium rule (in the cases I've tried)

ò₀^3h f(x) dx=h[f(0)+3f(h)+3f(2h)+f(3h)]/8 is exact on cubics f.

Both of the (Newton-)Simpson's rule and the Simpson's 3/8-rule are accurate up to cubics, and there is something called Gregory's formula that is a generalisation of these results. The fact that the (Newton-)Simpson's rule is exact for cubic is perhaps slightly surprising, but it is a consequence of ò_-h^h x³ dx=0.

Kerwin