Coding and speeding up a numerical integrator

Write an algorithm that numerically integrates a function over an interval, and explain how you would make it faster or more accurate.

Start with the trapezoid or Simpson rule, then refine adaptively. Split $[a,b]$ into $n$ panels and sum weighted function values: trapezoid error is $O(h^2)$, Simpson $O(h^4)$ for smooth integrands, each $O(n)$ evaluations. Optimise by halving the step and reusing previous points until successive estimates agree, Richardson/Romberg extrapolation, adaptive subdivision where the integrand varies, or Gaussian quadrature for smooth functions.