Question

How do I find the area inside a limacon?

Answer 1

In geometry, a limaçon, is defined as a roulette formed by the path of a point fixed to a circle when that circle rolls around the outside of a circle of equal radius. Limaçon curves look like circles. They have various types depending on the values in their equations. The polar equation of the limacon is $r=b+a\cos \theta$. We will assume that the limacon does not cross itself, for this condition to be true $\left| b \right|\ge \left| a \right|$. The infinitesimal segment of limacon has an area $\dfrac{1}{2}{{r}^{2}}d\theta$. To find the area of limacon, we have to integrate this over the range $0$ to $2\pi$.

Complete step by step solution:
We are asked to find the area inside a limacon, we know that the polar equation of a limacon is $r=b+a\cos \theta$. We know that the infinitesimal segment of limacon has an area $\dfrac{1}{2}{{r}^{2}}d\theta$. To find the area of limacon, we have to integrate this over the range $0$ to $2\pi$.
We can do this as follows,
$\int\limits_{0}^{2\pi }{\dfrac{1}{2}{{r}^{2}}d\theta }=\int\limits_{0}^{2\pi }{\dfrac{1}{2}{{\left( b+a\cos \theta \right)}^{2}}d\theta }$
Simplifying the above expression, we get
$\int\limits_{0}^{2\pi }{\dfrac{1}{2}{{r}^{2}}d\theta }=\int\limits_{0}^{2\pi }{\dfrac{1}{2}\left( {{b}^{2}}+{{a}^{2}}{{\cos }^{2}}\theta +2ab\cos \theta \right)d\theta }$
We can separate the integration over the addition of functions, thus we can simplify the above expression as
$\Rightarrow \dfrac{1}{2}\left( \int\limits_{0}^{2\pi }{{{b}^{2}}d\theta }+\int\limits_{0}^{2\pi }{{{a}^{2}}{{\cos }^{2}}\theta d\theta }+\int\limits_{0}^{2\pi }{2ab\cos \theta d\theta } \right)$
Integrating the above expression, we get
$\Rightarrow \dfrac{1}{2}\left( 2\pi {{b}^{2}}+\pi {{a}^{2}} \right)$
We can simplify the above expression to express it as
$\Rightarrow \pi \left( {{b}^{2}}+\dfrac{1}{2}{{a}^{2}} \right)$

Note: As we already said that limaçon curves look like circles. They have various types depending on the values in their equations. If the value of a in the polar equation of limacon is 0. Then, it becomes a special case that represents the circle. The radius of the circle is b. The area equation is also simplified as $\pi {{b}^{2}}$.