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Determine the total arc length of the cardioid r = 3 + 3\sin \theta

Question:

Determine the total arc length of the cardioid {eq}r = 3 + 3\sin \theta {/eq}

Find the Length of a Cardioid:

If {eq}r = f(\theta ) {/eq} is an equation of a cardioid then its length will be:

{eq}L = \int_0^{2 \pi} \sqrt{r^2 + ( \frac{dr}{d \theta })^2 } \, d \theta {/eq}

Where {eq}0 \leq \theta \leq 2 \pi {/eq}

Answer and Explanation:

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The equation of cardioid is given by:

{eq}r = 3 + 3 \sin (\theta ) {/eq}

Derivative of {eq}r {/eq} with respect to {eq}\theta {/eq} is:

{eq}\fra...

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Cardioid in Math: Definition, Equation & Examples

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Chapter 1 / Lesson 13
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This lesson will cover a neat shape studied in upper-level mathematics called a cardioid. We will look at the basic shape, how it is constructed, its equation in polar form, and various examples of these equations and corresponding cardioids.


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