Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 7 - Integration Techniques - 7.8 Improper Integrals - 7.8 Exercises: 60

Answer

\[ = 2 \times {10^9}\]

Work Step by Step

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