Use Taylor approximation on $\displaystyle e^{-x^{2}}$ to compute $\displaystyle \int_{0}^{1} e^{-x^{2}}\,dx$ to three decimal places and prove the accuracy of your answer using the theorem that if $\displaystyle f$ is of class $\displaystyle C^{k+1}$ on an interval $\displaystyle I$ and $\displaystyle |f^{(k+1)}(x)|le M$ for $\displaystyle x\in I$, then $\displaystyle |R_{a, k}(h)|\le \frac{M}{(k+1)!}|h|^{k+1}$.

So I know how to find the taylor expansion of $\displaystyle e^{-x^{2}}$ about zero and then integrate each term, but I do not know how to approximate to ensure that the approximation is good to three decimal places nor do I know how to apply that theorem to prove it. Any help would be appreciated.