Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 13: Vector-Valued Functions and Motion in Space - Section 13.2 - Integrals of Vector Functions; Projectile Motion - Exercises 13.2 - Page 753: 9


$$\mathbf{i}-\mathbf{j}+\frac{\pi}{4} \mathbf{k}$$

Work Step by Step

We evaluate the integral of the vector function as follows: \begin{align*} \int_{0}^{\pi / 2}\left[(\cos t) \mathbf{i}-(\sin 2 t) \mathbf{j}+\left(\sin ^{2} t\right) \mathbf{k}\right] d t&=\int_{0}^{\pi / 2}\left[(\cos t) \mathbf{i}-(\sin 2 t) \mathbf{j}+\left(\frac{1}{2}-\frac{1}{2} \cos 2 t\right) \mathbf{k}\right] d t \\ &=\sin t\bigg|_{0}^{\pi / 2} \mathbf{i}+ \frac{1}{2} \cos t\bigg|_{0}^{\pi / 2} \mathbf{j}+\left(\frac{1}{2} t-\frac{1}{4} \sin 2 t\right)\bigg|_{0}^{\pi / 2} \mathbf{k}\\ &=\mathbf{i}-\mathbf{j}+\frac{\pi}{4} \mathbf{k} \end{align*}
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