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: 39


Total length: $s \simeq 79.56$

Work Step by Step

We have $r = f\left( \theta \right) = \theta \sin \theta $, ${\ \ }$ $f'\left( \theta \right) = \sin \theta + \theta \cos \theta $. Using Eq. (7) the total length of the curve is $s = \mathop \smallint \limits_0^{4\pi } \sqrt {{{\left( {\theta \sin \theta } \right)}^2} + {{\left( {\sin \theta + \theta \cos \theta } \right)}^2}} {\rm{d}}\theta $ Evaluating it using a computer algebra system we obtain the result: $s \simeq 79.56$
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