First I showed the following:
$$\int_{0}^{1} \frac{x^4(1-x)^4}{1+x^2} \ dx \ = \frac{22}{7}-\pi$$
Now originally, we had to prove that
$$\frac{22}{7}-\frac{1}{630} \leq \pi \leq \frac{22}{7}-\frac{1}{1260}$$
But I was unable to figure out the upper bound $\frac{22}{7}-\frac{1}{1260}$ (I found that the lower bound can be found by integrating the numerator).
I later realised that we can use the fact that $\frac{x^4(1-x)^4}{2} \leq \frac{x^4(1-x)^4}{1+x^2}$ over the interval [0,1]. But this is not the main problem.
Initially, instead of using $\frac{x^4(1-x)^4}{2}$ I used the following inequality:
$$\frac{x^4(1-x)^4}{(1+x^2)^2} \leq \frac{x^4(1-x)^4}{1+x^2} \leq x^4(1-x)^4$$
Integrating both sides from 0 to 1:
$$\int_{0}^{1} \frac{x^4(1-x)^4}{(1+x^2)^2} dx \leq \int_{0}^{1} \frac{x^4(1-x)^4}{1+x^2} dx \leq \int_{0}^{1} x^4(1-x)^4 dx$$
Which simplifies to:
$$\frac{22}{7}-\frac{1}{630} \leq \pi \leq \frac{2}{7}(\frac{22}{7}+\ln{4}+\frac{97}{15}) \leq \frac{22}{7}-\frac{1}{1260}$$
I think I found a tigther upper bound, which gave me a better approximation of $\pi$
Also by using $\frac{x^n(1-x)^n}{1+x^2}$ with higher powers of n with the same upper and lower bounds, seemingly gave me better and better approximations of $\pi$ (I got 12 digits with 17th power, which is int(17/4) times the number of digits I got with 4th power).
But I'm not sure where I went wrong, because my professor said that higher powers will not give me more accurate values of $\pi$ and my inequalities are wrong. I checked by putting n = 17 and 27, but noticed that n = 17 gives me an lower bound instead of an upper bound (although still gives me more accurate values) if I use the inequality
$$\int_{0}^{1} \frac{x^{17}(1-x)^{17}}{(1+x^2)^2} dx \leq \int_{0}^{1} \frac{x^{17}(1-x)^{17}}{1+x^2} dx$$
whereas if I use n=27, i.e
$$\int_{0}^{1} \frac{x^{27}(1-x)^{17}}{(1+x^2)^2} dx \leq \int_{0}^{1} \frac{x^{27}(1-x)^{17}}{1+x^2} dx$$
I get an upper bound for the value of pi (which is accurate to about 15 digits) which is what I want.
The
$$\frac{x^n(1-x)^n}{1+x^2} \leq x^n(1-x)^n$$
inequality still gives me a lower bound that keeps getting better with higher powers of n.
What am I doing wrong here? (Sorry for the dumb questions - I'm not that smart)
