[yaakov34]

Even if you're limited to uniform sampling, something as simple as the trapezoid rule will give you quadratic convergence instead of linear for the naive Sum(f(t_i) * delta). In other words, error proportional to 1/n^2, instead of 1/n, where n is the number of samples, which is going to be a huge difference. There are many methods depending on the constraints of your problem - your ability to choose sampling intervals, knowledge of the bounds of your function or its derivatives, etc. The PSA is to study these things, instead of just writing the first thing that seems familiar from a long-ago calculus class.