How do you find the length of cardioid $r = 1 - cos theta$ ?

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Answer 1 · 2016-06-23T08:36:53

8

arc length in polar is
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here $(dr)/(d theta) = sin theta$

so $ds = sqrt{ sin^2 theta + (1-cos theta)^2 } \ d theta$

$= sqrt{ sin^2 theta + 1-2 cos theta + cos^2 theta } \ d theta$

$= sqrt(2) \ sqrt{ 1- cos theta } \ d theta$

$= sqrt(2) \ sqrt{ 1- (1 - 2 sin^2 (theta/2)) \ d theta$

$= sqrt(2) \ sqrt{ 2 sin^2 (theta/2)) \ d theta$

$= 2 \ sin (theta/2) \ d theta$

So, assuming length is of one full revolution.....
$S = 2 \ \int_0^{\color{red}{2 pi}} \ sin (theta/2) \ d theta$

$= 2 \ [ -2cos (theta/2) ]_0^{2 pi}$

$= 4 [ cos (theta/2) ]_{2 pi}^0$

$= 4 [ 1 - (-1) ] = 8$

How do you find the length of cardioid r = 1 - cos thetar = 1 - cos theta?