Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 9 - Further Applications of the Integral and Taylor Polynomials - 9.1 Arc Length and Surface Area - Preliminary Questions - Page 468: 2


See the proof below.

Work Step by Step

Applying the surface area formula and using the facts that $y=r$, $y'=0$, we have the integral, $$2\pi\int_0^{h}r\sqrt{1+(y')^2}dx=2\pi\int_0^{h}rdx=2\pi r x|_0^h=2\pi r h.$$ Which is the surface area of a cylinder of radius r and length h.
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