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Answers without the blur. Sign up and see all textbooks for free! Q. 33

Expert-verified Found in: Page 756 ### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861 # in exercise 31-36 find a definite integral that represents the length of the specified polar curve, and then use graphing calculator or computer algebra system to approximate the value of integral One petal of the polar rose $r=\mathrm{cos}4\theta$

The integral can be given as $2{\int }_{0}^{\frac{\pi }{8}}\sqrt{{\mathrm{cos}}^{2}4\theta +16{\mathrm{sin}}^{2}4\theta }d\theta$ and the arc length can be given as 2.14units.

See the step by step solution

## Step 1: Given information

We are given the equation as $r=\mathrm{cos}4\theta$

## Step 2: Evaluate

We know that the arc length of one petal of the polar rose $r=\mathrm{cos}\left(n\theta \right)$ can be given as

$2{\int }_{0}^{\frac{\pi }{2n}}\sqrt{{\mathrm{cos}}^{2}n\theta +{n}^{2}{\mathrm{sin}}^{2}\theta }d\theta \phantom{\rule{0ex}{0ex}}Putn=4weget,\phantom{\rule{0ex}{0ex}}2{\int }_{0}^{\frac{\pi }{8}}\sqrt{{\mathrm{cos}}^{2}4\theta +16{\mathrm{sin}}^{2}4\theta }d\theta$

Using a CAS calculator we get,

$L=2\left(1.07\right)\phantom{\rule{0ex}{0ex}}L=2.14unit$ ### Want to see more solutions like these? 