Calculus: Early Transcendentals (2nd Edition)

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

Chapter 5 - Integration - Review Exercises: 23

Answer

\[ = \frac{\pi }{6}\]

Work Step by Step

\[\begin{gathered} \int_0^1 {\frac{{dx}}{{\sqrt {4 - {x^2}} }}} \hfill \\ \hfill \\ rewrite\,\,the\,\,deno\min ator \hfill \\ \hfill \\ = \int_0^1 {\frac{{dx}}{{\sqrt {\,{{\left( 2 \right)}^2} - {x^2}} }}} \hfill \\ \hfill \\ use\,\,\,\,the\,\,formula\,\int_{}^{} {\frac{{du}}{{\sqrt {{a^2} - {u^2}} }}} \, = \,\,\arcsin \,\left( {\frac{u}{a}} \right) + C \hfill \\ \hfill \\ = \,\,\left[ {\arcsin \,\left( {\frac{x}{2}} \right)} \right]_0^1 \hfill \\ \hfill \\ Fundamental\,\,theorem \hfill \\ \hfill \\ = \arcsin \,\left( {\frac{1}{2}} \right) - \arcsin \,\left( 0 \right) \hfill \\ \hfill \\ = \frac{\pi }{6} \hfill \\ \end{gathered} \]
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