Answer
$3$
Work Step by Step
\[\begin{align}
& f\left( x \right)=3{{e}^{-x}},\text{ }\left[ 0,\infty \right) \\
& \text{The area is given by} \\
& A=\int_{0}^{\infty }{3{{e}^{-x}}}dx \\
& \text{Using the definition of improper integrals} \\
& A=\underset{b\to \infty }{\mathop{\lim }}\,\int_{0}^{b}{3{{e}^{-x}}}dx \\
& A=\underset{b\to \infty }{\mathop{\lim }}\,\left[ -3{{e}^{-x}} \right]_{0}^{b} \\
& A=\underset{b\to \infty }{\mathop{\lim }}\,\left[ -3{{e}^{-b}}+3{{e}^{0}} \right] \\
& A=\underset{b\to \infty }{\mathop{\lim }}\,\left[ -3{{e}^{-b}}+3 \right] \\
& \text{Evaluate when }b\to \infty \\
& A=-3{{e}^{-\infty }}+3 \\
& A=0+3 \\
& A=3 \\
\end{align}\]
