The most accurate method
of approximating the area under a curve when it is impossible to integrate
Simpson, in his infinite wisdom, theorized that it's better to approximate areas under curves using curves rather than rectangles. So, he used the most basic curve (and decidedly easiest to integrate!)- the parabola.
He split the curve-in-question up into 3 points and drew a parabola using the 3 coordinates. The results were much more accurate than using left/right hand rectangles, trapezoids, and even midpoint rectangles.
So, it came to pass that the Simpson rule or "Simpson's method" be used as a standard in approximating areas under curves. The only disadvantage to using Simpson's method is that one must have an even number of terms to plug in. (ie: you can't do the interval of 1,8 with 7 sub-intervals.)
The formula is as follows (for function f from a to b, step n):
SIMP = (δx/3) * ( f(a) + 2 * f(a + n) + 4 * f(a + 2n) ... + f(b) )