Answer
\[ = 2 \times {10^9}\]
Work Step by Step
\[\begin{gathered}
Rate\,\,of\,\,extracting\,\,oil\,\,is\,\,\,given\,\,by \hfill \\
\hfill \\
r\,\left( t \right) = {r_0}{e^{ - kt}} \hfill \\
\hfill \\
where \hfill \\
\hfill \\
{r_0} = \frac{{{{10}^7}\,barrels}}{{yr}} \hfill \\
\hfill \\
and \hfill \\
\hfill \\
k = 0.005\,\,y{r^{ - 1}} \hfill \\
\hfill \\
we\,\,\,the\,\,estimate\,\,of\,\,the\,\,total\,\,oil\,\,reserve\,\,is \hfill \\
2 \times {10^9}\,barrels. \hfill \\
\hfill \\
the\,\,extraction\,\,continues\,\,indefinitely,\,\,so \hfill \\
\hfill \\
\int_0^\infty {r\,\left( t \right)} \,dt\,\,\, = \,\,\,\int_0^\infty {{r_0}{e^{ - kt}}} \,dt \hfill \\
\hfill \\
{\text{integrate}}\,\,{\text{using}}\,\,{\text{the}}\,\,{\text{definiton}}\,\,{\text{of}}\,\,{\text{improper}}\,\,\,{\text{integrals}} \hfill \\
\hfill \\
= \mathop {\lim }\limits_{b \to \infty } \,\,\int_0^b {{{10}^7}{e^{ - 0.005t}}} \,dt \hfill \\
\hfill \\
= \frac{{{{10}^7}}}{{ - 0.005}}\,\,\mathop {\lim }\limits_{b \to \infty } \,\,\left[ {{e^{ - 0.005b}} - 1} \right] \hfill \\
\hfill \\
{\text{evaluate}}\,\,{\text{the}}\,\,{\text{limit}} \hfill \\
\hfill \\
= \frac{{{{10}^7}}}{{0.005}} = 2 \times {10^9} \hfill \\
\end{gathered} \]
