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