Answer
$$1$$
Work Step by Step
$$Area=\int_{0}^{\infty} e^{-x} dx \\=\lim\limits_{a \to \infty}\int_{0}^{a} e^{-x} dx \\=\lim\limits_{a \to \infty}[-e^{-x}]_{0}^a \\=\lim\limits_{a \to \infty}[-e^{-a}-(-e^{-1})] \\=0-(-1)=0+1\\=1$$
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