Prove that $\displaystyle \frac{\pi}{2} - 1 < \int^1_0 e^{-2x^2}\,\mathrm{d}x.$

I thought of integrating the inequality $\displaystyle 2\sqrt{1-x^2} - 1 \leq e^{-2x^2}$ from 0 to 1 but the inequality isn't strict. I'm not sure what I could do to make it strict; I think a different approach is needed.

Any ideas? Thanks.