I am curious about the value of Simpson's rule (also called the parabolic rule or the 3-point rule) for approximating integrals. The calculus text I am now teaching from uses this rule any time an approximation is needed for an integral. For example, it may give a messy arclength integral and ask for the Simpson's rule approximation using 4 intervals (and thus 5 sample points):
$$ \int_a^{a+4h} f(x) dx \simeq \frac{h}{3}\left(f(a)+ 4f(a+h) + 2f(a+2h) + 4f(a+3h)+ f(a+4h)\right).$$
I understand the idea of Simpson's Rule. If you just sampled three evenly spaced points on a quadratic function, you could compute the integral on that interval with the weighting pattern $(1,4,1)$; this happens to give the right answer for degree 3 polynomials as well. The $(1,4,2,4,2, \ldots, 4,2,4,1)$ pattern comes from repeating this pattern over every pair of intervals.
But I'm not convinced we should always apply this rule any time we cut into $2n$ intervals. Why not just use throw out the uneven weighting and use a few more sample points? If the weighting is so helpful, why not use a more complicated weighting (like the various $n$-point rules (Newton-Cotes formulas) described here)?
The Newton-Cotes formulas and their error terms form a beautiful theory, but are probably too much for undergrad calculus! I understand showing Simpson's rule and going no further.
So I have two main questions-- is Simpson's rule so useful that calculus students should always use it for approximations? And are the other Newton-Cotes formulas (or Gaussian quadrature) always the best way to do numerical integration, or only when the values $f(x_i)$ are sufficiently expensive to compute?