Thomas' Calculus 13th Edition

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

Chapter 16: Integrals and Vector Fields - Section 16.5 - Surfaces and Area - Exercises 16.5 - Page 989: 21


$$6 \space \pi $$

Work Step by Step

Apply cylindrical coordinates. $ x=r \cos \theta ;\\ y= r \sin \theta ;\\ z \gt 0$ We know that $ r(r, \theta)=xi+yj+zk $ or, $ r^2=x^2+y^2+z^2$ We have $ z=1; z=4$ and $ x^2+y^2=1$ Now, $ r_z=\cos \theta \space i+ \sin \theta \space j+2 \space k; \\ r_{\theta}=-\sin \theta \space i+ \cos \theta \space j $ Also, $|r_z \times r_{\theta}|=1$ Now, $$ Area =\int_0^{2 \pi} \int_1^{4} (1) \space dr d \theta \\=\int_0^{2 \pi} 3 \space d \theta=6 \space \pi $$
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