22/7, an overestimate
The first approximation anyone learns is $22/7 = 3.142857\ldots$ It errs high, and the proof of that fact fits in one display line:
$$0 \;<\; \int_0^1 \frac{x^4(1-x)^4}{1+x^2}\,dx \;=\; \frac{22}{7}-\pi.$$Positivity is immediate: the integrand is continuous, nonnegative on $[0,1]$, and not identically zero, so the integral is strictly positive.1 The evaluation is nothing deeper than polynomial division:
1
and integrating term by term over $[0,1]$ gives $\tfrac17-\tfrac23+1-\tfrac43+4-\pi=\tfrac{22}{7}-\pi$. That the answer lands exactly on the schoolroom fraction is the charm of the thing.
How wrong is 22/7
The same integrand measures its own error. On $[0,1]$ we have $1\le 1+x^2\le 2$, so the integral is squeezed between half and all of
$$\int_0^1 x^4(1-x)^4\,dx \;=\; B(5,5) \;=\; \frac{4!\,4!}{9!} \;=\; \frac{1}{630},$$which yields
$$\frac{1}{1260} \;<\; \frac{22}{7}-\pi \;<\; \frac{1}{630}.$$So $22/7$ overshoots by about $0.00126$ — wrong in the fourth decimal, exactly as observed. Archimedes, squeezing the circle between 96-gons, already had $223/71 < \pi < 22/7$; the integral turns the upper half of his sandwich into a one-line calculus exercise.
The trick generalizes. Heavier integrands of the same family witness the next famous convergent, $355/113$, with its astonishing seven-digit accuracy.2 The digits of $\pi$ start at this article’s own address: the catalog places each constant at its leading digits, and $3.141\ldots$ lives at ℵ3141.