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 990: 40



Work Step by Step

Apply cylindrical coordinates. $ x=r \cos \theta ;\\ y= r \sin \theta ;\\ z \gt 0$ We know that $ x=y^2$ and $ x=2-y^2$ $$ Surface \space Area =\iint_{R} \dfrac{|\nabla f|}{|\nabla f \cdot p|} \space dA\\=\int_{0}^{\sqrt 3} \int_{0}^{x}\sqrt {x^2+1} \space dx \space dy \\ =(1/3) \times [(x^2+1)^{3/2}]_{0}^{\sqrt 3} \space \\=\dfrac{7}{3}$$
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