Hello, Gracy!
Find the area of the region that lies inside circle $\displaystyle r=1$
and outside the cardioid $\displaystyle r\:=\:1\cos x$
Sketch the graph. Code:

o o 
o * o *
o *  o::*
o *  o::*
o *  o:::*
 o:::::
o *  o::::::*
o*o:::::::::*
o *  o::::::*
 o:::::
o *  o:::*
o *  o::*
o *  o *
o * o *
o o 

The curves intersect when: .$\displaystyle 1 \:= \:1  \cos x$
. . Then: .$\displaystyle \cos x \,= \,0\quad\Rightarrow\quad x\:=\:\frac{\pi}{2},\:\frac{\pi}{2}$
Due to the symmetry, we can integrate from $\displaystyle 0$ to $\displaystyle \frac{\pi}{2}$ and double.
The shaded area is: .(Area of the circle)  (Area of the cardiod)
Therefore: .$\displaystyle A \;=\;2 \times \frac{1}{2}\int^{\frac{\pi}{2}}_0\left[1^2  (1  \cos x)^2\right]\,dx$
Go for it!