Calculus: Early Transcendentals (2nd Edition)

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

Chapter 5 - Integration - 5.5 Substitution Rule - 5.5 Exercises: 41

Answer

\[ = \frac{1}{3}\]

Work Step by Step

\[\begin{gathered} \int_0^{\frac{\pi }{2}} {{{\sin }^2}\theta \cos \theta d\theta } \hfill \\ \hfill \\ u = \sin \theta \,\,\,\,\,\,\,then\,\,\,\,du = \cos \theta d\theta \hfill \\ \hfill \\ {\text{Changing limits of integration}} \hfill \\ \hfill \\ \theta = 0\,\,\,\,\,\,implies\,\,u = 0 \hfill \\ \theta = \frac{\pi }{2}\,\,\,\,implies\,\,u = 1 \hfill \\ \hfill \\ apply\,\,the\,\,\,substitution \hfill \\ \hfill \\ = \int_0^1 {{u^2}\,du} \hfill \\ \hfill \\ integrate \hfill \\ \hfill \\ = \,\,\left[ {\frac{{{u^3}}}{3}} \right]_0^1 \hfill \\ \hfill \\ Fundamental\,\,theorem \hfill \\ \hfill \\ = \frac{1}{3} - 0\, = \frac{1}{3} \hfill \\ \hfill \\ \end{gathered} \]
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