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

Answer 1

Elijah Abram

arc length in polar is

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 #

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Answer 2

Luke Alvarez

To find the length of the cardioid ( r = 1 - \cos(\theta) ), you can use the formula for arc length in polar coordinates:

[ L = \int_{\alpha}^{\beta} \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2} , d\theta ]

where ( \alpha ) and ( \beta ) are the limits of integration that cover the desired portion of the curve.

For the cardioid ( r = 1 - \cos(\theta) ), you need to find ( \frac{dr}{d\theta} ) and then integrate using the appropriate limits.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.