My favorite integral aka I don’t care what your teacher told you, π ≠ 22/7.
By calculating a very simple integral we can see that π ≠ 22/7 regardless to what we were told in school.
I constantly have students coming into my college classes and trying to interchange between π and 22/7. Let’s be clear, 22/7 is a relatively good approximation for π (it is correct to 2 decimal places) but it certainly is not equal to π. I want to show you an integral I like to get my class to calculate at the end of an ``Introduction To Calculus Course” to exemplify the fact that these numbers are not equal. Consider the following integral
This may look somewhat difficult, but it actually works out to be quite simple to solve. I came across this problem when I ran the training for my University's math society, yes, I was that guy! It, or at least a version of it, appeared in a Putnam Competition. For those who don't know what the Putnam Competition is, very briefly, it is a VERY difficult exam. The exam, which is marked out of 120 points, has an average score between 2 and 8 points. So yeah, not to be taken lightly. It is worth noting that, generally speaking, the problems in the Putnam Competition are not "hard", they just require very VERY good insight into solving mathematical problems.
Anyway, I am going to solve this integral in what I believe to be the most pedestrian way. By this I mean that I am not going to use any clever tricks or substitutions, but instead will base my solution off methods that anyone who has taken any level of Calculus course should be able to follow. The first thing I will do is to convert the function (x⁴(1-x)⁴)/(1+x²) into a polynomial (with a remainder) and thus simplify my integral. To do this I will use the binomial theorem to express (1-x)⁴ as a simple polynomial.
For those who know the binomial theorem, great, for those who don't just take my word that it can be shown that
From this we can see that
We now want to divide this new expression by 1+x² to obtain a new expression for our original function. We can do this by polynomial long division. You can check for yourself that this works out as follows:
In particular, we find that
Thinking back to your calculus classes you might recognize this final integral:
The final thing to note is that our integral looked at the function (x⁴(1-x)⁴)/(1+x²) for x between 0 and 1. But, for x ∈ (0,1),
In particular, all of these different parts of our function are strictly bigger than zero and so,
Thus, we must have that
For those of you who have studied, or will go onto to study, “continued fractions” you will see there is a very natural reason as to why 22/7 is taken as the approximation of π in schools. You will also see that the method used to generate this approximation will also generate, with very little extra work, much much (yes I meant to write much twice there) better approximations to π.