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Let \(f\) be the function defined on \(\mathbb{R}\) by \(f(x) = e^{-x^2}\).
  1. Show that for all real numbers \(x \ge 1\), \(0 \le f(x) \le e^{-x}\).
  2. Deduce a bound for the integral \(\displaystyle \int_{1}^{2} f(x) \,dx\).

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