Simpson's rule

Trapezoids join the points with straight segments. But a curve is… curved. Hence the brilliant idea behind Simpson's rule: join the points with small arcs of parabolas, which follow the curvature.

We group the points three at a time — so we need an even number of pieces — and, on each pair of pieces, we pass the parabola that agrees with $f$ at those three points. The area under these parabolas gives a remarkable approximation:

$$\int_a^b f(x)\,\mathrm{d}x \approx \frac{\Delta x}{3}\Bigl[f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \cdots + 4f(x_{n-1}) + f(x_n)\Bigr]$$

The pattern of coefficients is distinctive: $1, 4, 2, 4, 2, \ldots, 4, 1$. The endpoints count $1$, the points with odd index count $4$, and the interior points with even index count $2$.

Method	What approximates the curve	Order of the error
rectangles	flat top	$\dfrac{1}{n}$
trapezoids	straight segment	$\dfrac{1}{n^2}$
Simpson	arc of a parabola	$\dfrac{1}{n^4}$

Simpson's accuracy is spectacular: the error decreases like $\dfrac{1}{n^4}$. Better still, the method gives the exact result for every polynomial of degree $\leq 3$. On $f(x) = x^2$ with only $4$ pieces, Simpson already finds the exact value $\tfrac{1}{3}$, whereas the trapezoids are still off:

Method, with $n = 4$	$\displaystyle\int_0^1 x^2\,\mathrm{d}x$
trapezoids	$\approx 0.34375$
Simpson	$0.33333$, exact value

Remember the progression: flat, then straight, then curved. Each level fits $f$ better, and accuracy shoots up for the same number of points. This is why Simpson's rule is so widely used in practice.