Calculus (3rd Edition)

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

Chapter 12 - Parametric Equations, Polar Coordinates, and Conic Sections - 12.4 Area and Arc Length in Polar - Exercises - Page 625: 40


Total length: $s \simeq 30.50$

Work Step by Step

We have $r = f\left( \theta \right) = \sqrt \theta $, ${\ \ }$ $f'\left( \theta \right) = \frac{1}{{2\sqrt \theta }}$. Using Eq. (7) the total length of the curve is $s = \mathop \smallint \limits_0^{4\pi } \sqrt {{{\left( {\sqrt \theta } \right)}^2} + {{\left( {\frac{1}{{2\sqrt \theta }}} \right)}^2}} {\rm{d}}\theta = \mathop \smallint \limits_0^{4\pi } \sqrt {\theta + \frac{1}{{4\theta }}} {\rm{d}}\theta $ Evaluating it using a computer algebra system we obtain the result: $s \simeq 30.50$
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