2009/017) A cute little integration to find the lower and upper bound for pi

Integrate

x^4(1-x)^4/(1+x^2) from 0 to 1

expanding we get

x^6-4x^5+5x^4-4x^2+4-[4/(1+x^2)]

integrating we get

x^7/7 – 2/3x^6+ x^5- 4/3x^2 + 4x – 4tan ^-x

x=1 gives 1/7-2/3+1-4/3 + 4 = 22/7 – 4 arctan(1) = 22/7- pi

x=0 gives 0

so definite integral = 22/7 – pi

now as the LHS is positive at each point integral > 0 so 22/7 –pi or pi < 22/7

now for the lower limit let us find the higher limit of LHS

x(1-x) is highest at x= ½ and x(1-x) = ¼

so x^4(1-x)^4 highest is 1/256

and lowest of (1+x^2) is 1

so integral of LHS < 1/256

so 22/7-1/256 < pi < 22/7

gives pi(which is 3.14159..) between 3.1389 and 3.142857

pretty good is it not ?