Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 13 - Vector Functions - 13.2 Derivatives and Integrals of Vector Functions - 13.2 Exercises: 35

Answer

$2\mathbf{\hat{i}}-4\mathbf{\hat{j}}+32\mathbf{\hat{k}}$

Work Step by Step

1) Anti-differentiate and evaluate $\int_{0}^{2} (t\mathbf{\hat{i}}- t^3\mathbf{\hat{j}}+ 3t^5\mathbf{\hat{k}})dt$ using power rule. $$\left[\frac{1}{2}t^2\mathbf{\hat{i}}- \frac{1}{4}t^4\mathbf{\hat{j}}+ \frac{1}{2}t^6\mathbf{\hat{k}}\right]_{0}^{2}\\ =\left[\frac{1}{2}(2)^2\mathbf{\hat{i}}- \frac{1}{4}(2)^4\mathbf{\hat{j}}+ \frac{1}{2}(2)^6\mathbf{\hat{k}}\right]- \left[\frac{1}{2}(0)^2\mathbf{\hat{i}}- \frac{1}{4}(0)^4\mathbf{\hat{j}}+ \frac{1}{2}(0)^6\mathbf{\hat{k}}\right]\\ =2\mathbf{\hat{i}}-4\mathbf{\hat{j}}+32\mathbf{\hat{k}} $$
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