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**My attempt:** $E_n^T(f)\approx -h^2/12[f'(b)-f'(a)]. f(x)=e^{-x^2}, f'(x)=-2xe^{-x^2}.$

So $E_n^T(f)\approx h^2/(4e^4)$ and since $h=1/n$, we have to find an n that satisfies the inequality $\frac{1}{4e^4n^2}\leq 10^{-10}$. We obtain $n \approx 6767$. The approximate integral is ??