# Question e8898

Feb 2, 2017

The common points are $\left(\frac{3}{2} , - \frac{\pi}{3}\right) \mathmr{and} \left(\frac{3}{2} , \frac{\pi}{3}\right)$.
Area inside the circle and outside the cardioid is $9 \left(\sqrt{3} - \frac{\pi}{3}\right)$
= 6.16368 areal units, nearly. See graph..

#### Explanation:

At the common points of the circle

$r = 3 \cos \theta$ and the cardioid

$r = 3 \left(1 - \cos \theta\right)$,

$r = 3 \cos \theta = 3 \left(1 - \cos \theta\right)$, giving $\cos \theta = \frac{1}{2}$

In $\left(- \pi , \pi\right)$, the angles $\alpha \mathmr{and} \beta$ as solutions of this

equation are $\pm \frac{\pi}{3}$.

The area A inside the circle and outside the cardioid is symmetrical

about $\theta = 0$. So,

A = 2 (1/2int r^2 d theta#, for the circle

$- \frac{1}{2} \int {r}^{2} d \theta$. for the cardioid),

with $\theta$ from 0 to $\frac{\pi}{3}$

$= 9 \int \left({\cos}^{2} \theta - {\left(1 - \cos \theta\right)}^{2}\right) d \theta$, for the limits

$= 9 \int \left(2 \cos \theta - 1\right) d \theta$, for the limits

$= 9 \left[2 \sin \theta - \theta\right]$, between $0 \mathmr{and} \frac{\pi}{3}$

$= 9 \left(\sqrt{3} - \frac{\pi}{3}\right)$

graph{(x^2+y^2-3x)(x^2+y^2-3sqrt(x^2+y^2)+3x)=0x^2 [-10, 10, -5, 5]}