The most accurate integral for the value of pi

[stef.grob]

Hey all we doing some work with integrals and Our lecturar mentioned that one can get a answer for pi using integrating, my question is what would that integral be, and wot is the most accurate? I've gotten it till 3 decimal places.

 

[Hurkyl]

Floating-point algorithms for computing pi are usually infinitely accurate and precise -- if you give the algorithm a tolerance a, it will give you a floating-point number within the interval (pi-a, pi+a).

It's just that if the tolerance is small, it takes a lot of work.

This is actually one of the things you learn in calculus class. You know, for example:
$$\int_0^1 \sqrt{1 - x^2} \, dx = \frac{\pi}{4}$$

If you wanted to apply Simpson's rule to this integral to compute, say, 5 decimal digits of pi, there's a theorem you can invoke that will tell you how many terms you need to use. You have to take extra care, because you will probably introduce extra error in your calculation -- your calculus class only gives you a brief introduction to the field of numerical calculation.



I don't know of any integral-based approximations that are efficient, though. The fastest algorithms tend to be based on infinite sums.

 

[g_edgar]

I don't know of any integral-based approximations that are efficient, though. The fastest algorithms tend to be based on infinite sums.

On the other hand, I don't know a proof that pi is irrational based on a sum for pi, only a proof based on an integral.