Hello, s7b!
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Find the area of the region common to the interiors of the cardioids;
. . $\displaystyle r\:=\:1+\cos\theta\,\text{ and }\,r\:=\:1\cos\theta$ Code:

*  *
* *  * *
*
* *::* *
*::::*
* *::* *
  *           *           *  
* *::* *
*::::*
* *::* *
*
* *  * *
*  *

Then intersect when: .$\displaystyle 1 + \cos\theta \:=\:1\cos\theta \quad\Rightarrow\quad \theta \:=\:\pm\tfrac{\pi}{2} $
The region has fourway symmetry.
We can find the area in Quadrant 1 and multiply by 4.
The integral is: /$\displaystyle \text{Area} \;=\;4 \times\tfrac{1}{2}\int^{\frac{\pi}{2}}_0\left(1  \cos\theta\right)^2d\theta $