Calculus Concepts: An Informal Approach to the Mathematics of Change 5th Edition

Published by Brooks Cole
ISBN 10: 1-43904-957-2
ISBN 13: 978-1-43904-957-0

Chapter 5 - Accumulating Change: Limits of Sums and the Definite Integral - 5.9 Activities - Page 409: 23


$a)\int ^{5}_{0}\left( \sin x\right) ^{2}\cos xdx=\dfrac {\left( \sin x\right) ^{3}}{3}+c$$ $$b)\int ^{5}_{0}\left( \sin x\right) ^{2}\cos xdx=\dfrac {\left( \sin 5\right) ^{3}-\left( \sin 0\right) ^{3}}{3}\approx -0.29$

Work Step by Step

$u=\sin x\Rightarrow du=\cos xdx$ $$a)\int ^{5}_{0}\left( \sin x\right) ^{2}\cos xdx=\int ^{5}_{0}u^{2}du=\dfrac {u^{3}}{3}+c=\dfrac {\left( \sin x\right) ^{3}}{3}+c$$ $$b)\int ^{5}_{0}\left( \sin x\right) ^{2}\cos xdx=\dfrac {\left( \sin x\right) ^{3}}{3}]^{5}_{0}=\dfrac {\left( \sin 5\right) ^{3}-\left( \sin 0\right) ^{3}}{3}\approx -0.29$$
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