The most

accurate method of approximating the area under a curve when it is impossible to

integrate the

function.

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) )