using integration to approximate irrational numbers

Hello,

In the "caclulus tutorial" thread posted at the top of this forum, TPH (the author of the thread) mentions how he came up with an integral that equaled . Then he says this equation is useless precisely because it equals exactly, that what is needed is an equation that approximates it. Can someone please explain why you would want an equation that approximates the solution rather than gives an exact solution?

For reference the exact quote is below.

TPH:

"When I was younger the method I used determine the value of http://www.mathhelpforum.com/math-he...7a9dac6d-1.gif was based on the area below a circle. That is, http://www.mathhelpforum.com/math-he...d4184689-1.gif is a semicircle with radius of 2. Thus, the area of quater circle is http://www.mathhelpforum.com/math-he...7a9dac6d-1.gif that is, http://www.mathhelpforum.com/math-he...4a49ab63-1.gif, thus, I was all excited that I discovered an equation that solves for http://www.mathhelpforum.com/math-he...7a9dac6d-1.gif, but the funny thing is that if you actually evaluate that integral (by using advanced techniques that we did not discuss) you will get http://www.mathhelpforum.com/math-he...7a9dac6d-1.gif, thus you get http://www.mathhelpforum.com/math-he...d3e91d4f-1.gif which gives you nothing. However, you can approximate that integral by approximation methods (which we also did not discuss but I can show the general idea of how it is done). And hence we get an equation that approximates http://www.mathhelpforum.com/math-he...7a9dac6d-1.gif, (though it can be shown the method is not very efficient). Thus, we have shown how integration can be used in approximation of some irrational numbers, and how mathematicians do it."