Past exam of the year 1 of the computer science course of the University of Oxford / 2016 / Discrete mathematics, continuous mathematics and probability / 6 / b

Set $h=(b-a)/n$ and $x_i=a+ih$. Summing the areas of the $n$ endpoint trapezia gives the composite trapezoidal ruleEvery interior value is shared by two adjacent strips; the two endpoint values occur only once with half weight. Evenness of $n$ is not needed for this rule.

For the same uniform grid and even $n$, the composite Simpson's rule integrates a quadratic interpolant over each successive pair of strips:The weights are $1,4,2,4,\dots ,2,4,1$, which explains the even number of strips.