Looking at an expression for $\pi$
$$\pi = \frac{22}7-\int_0^1 \frac{x^4(1-x)^4}{1+x^2} \ dx$$
it seems to me that the integral expression is the error between the approximation $\frac{22}7$ and $\pi$. Is there any history behind the discovery of this integral? Was it tied to the error between the two numbers or was it discovered separately and only later applied?
This is the earliest instance of the formula that I could find.
http://jlms.oxfordjournals.org/content/19/75_Part_3/133
J. London Math. Soc. (1944) 19 (75 Part 3): 133-134. doi: 10.1112/jlms/19.75_Part_3.133
Perhaps this is where it was first discovered.