Use your previously implemented function for Monte Carlo integration.

1. Find out how large the sample must be to approximate the integral $\displaystyle\int_0^1\frac{4}{x^2+1}dx$ with a precision of $0.1$, $0.01$, $0.001$ and $0.0001$.
   
   
2. Compare the result from part (a) with the following probability-based algorithm to approximate an integral $\displaystyle \int_a^bf(x)\,\dd x$: draw a uniform sample of values in the interval $[a,b]$, calculate the mean value of the function $f$ for the drawn values in the sample and multiply this mean value by the length of the interval, i.e. by $b-a$.