Answer
$$
S=\left\{(x, y) | x \geqslant 1,0 \leqslant y \leqslant e^{-x}\right\}
$$
The shaded area is the region of interest.
$$
\begin{split}
\text { Area } & = \int_{1}^{\infty } e^{-x} d x \\
& =1 / e
\end{split}
$$
Work Step by Step
$$
S=\left\{(x, y) | x \geqslant 1,0 \leqslant y \leqslant e^{-x}\right\}
$$
The shaded area is the region of interest.
$$
\begin{split}
\text { Area } & = \int_{1}^{\infty } e^{-x} d x \\
& = \lim _{t \rightarrow \infty } \int_{1}^{ t } e^{-x} d x \\
& =\lim _{t \rightarrow \infty}\left[-e^{-x}\right]_{1}^{t} \\
& =\lim _{t \rightarrow \infty}\left(-e^{-t}+e^{-1}\right)=0+e^{-1}=1 / e
\end{split}
$$
