We can estimate the numerical value of $\displaystyle \int_{0}^{\,\infty} {e^{-x^2}}\, dx$ by writing it as the sum of $\displaystyle \int_{0}^{\,4} {e^{-x^2}}\, dx$ and $\displaystyle \int_{4}^{\,\infty} {e^{-x^2}}\, dx.$ It is possible to show that $\displaystyle \int_{4}^{\,\infty} {e^{-x^2}}\, dx$ is less than $\displaystyle \int_{4}^{\,\infty} {e^{-4x}}\, dx,$ which is less than 0.0000001.

Approximate the value of $\displaystyle \int_{0}^{\,4} {e^{-x^2}}\, dx$ by using Simpson's Rule with $n = 8$. Make sure that your answer is correct to four decimal places.