Chapter 11.6, Problem 29E

To determine

To obtain: An approximation of $\pi$ by using an eighth-degree Taylor polynomial of $f\left(x\right)=\frac{1}{1+x^2}$ at $x=0$ and the relationship $\pi =4{\displaystyle \underset{0}{\overset{1}{\int }}\frac{dx}{1+x^2}}$.