So for the functions I've tried over the intervals I've tried, this formula in blue lets you approximate the integral over an interval using just 5 carefully chosen points and in just a few additions, subtractions and multiplications to get at what before took an infinite number of them. And it gives an exact answer whereas you could only keep getting closer before.

*UPDATE*

Using this formula A(1) +(4/3)[A(2)-A(1)]

to evaluate integral(e^x, x=0..1) gives 1.717776531 instead of the true value of 1.718281828 so it's off a little for exponentials . Seems to work for polynomials though.

*UPDATE2*

It turns out that when it works you only need 3 points' y values to find the integral, over your interval [a,b], you need to know f((b-a)/2), and f of both halfway points between that and the endpoints. Let's say our interval was [2,7] and we measured those 3 points to be f(4.5)=-50.625, f(3.25)=-13.203125,

f(5.75)=-123.984375 after applying the blue formula:

we get: -4475/12

I hid from the formula but I know that these 3 points correspond to -x^3+2*x^2 the integral over that interval of which is:

-4475/12

Using the formula for x^4 +x^2 was off by 7.46, I get 658.666 instead of 666.1333

Still pretty close but I guess it's only exact up to cubics.

*Note That gives me the idea that if you know n points of a given curve, you can find the nth degree polynomial that runs through those (there's a formula for that) then integrate that over your boundaries, this should still be much faster than using an infinite number of rectangles.

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