There’s a reasonably well-known proof that , which can be written in one line:
But I’ve always found this one unsatisfying because what does that integral have to do with anyway? As it turns out, enters through the integral . But let’s say I’m a purist and think that is about circles. Can I do better? (Of course I can. If I couldn’t I wouldn’t be writing this post.) Start by observing that , which can be shown by explicit computation: and .
Edited to add, November 30: of course I just showed here that . This is what happens when you do arithmetic in your head…
Now, the sine function is given by the alternating series and in particular by the alternating series test. Applying this with gives .
Alejandro Morales, Igor Pak, and Greta Panova, Why is π < 2φ? (This is not a particularly good approximation, but it actually admits a combinatorial proof in terms of Fibonacci and Euler numbers.)
God plays dice 
Nice little proof. Their must be a problem with your TeX code. The integral is not right. Is there a missing ‘\’ before your over?
\int_0^1 \frac{x^4(1-x)^4}{1+x^2} \, dx = \frac{22}{7} – \pi
looks to be correct code, stolen from https://en.wikipedia.org/wiki/Proof_that_22/7_exceeds_π. I hope wordpress does not mangle the TeX I posted.
The proof is actually incorrect, (21/11)^3 is not greater than 7.