Question

Find the area common to the 2 regions bounded by:
r = - 6 cos(x) 
r = 2 - 2 cos(x) 

Answer 1

First area of the cardioid
\[
\text{area3}=\int_0^{2 \pi }
\frac{1}{2} (2-2 \cos
(x))^2 \, dx= 6\pi
\]

Answer 2

Then the area outside the circle but within the cardioid
\[
2 \left(\int_0^{\frac{2 \pi
}{3}} \frac{1}{2} (2-2 \cos
(x))^2 \, dx-\int_{\frac{2
\pi }{3}}^{\frac{\pi }{2}}
\frac{36 \cos ^2(x)}{2} \,
dx\right)=\\2\left ( 2 \pi -\frac{9 \sqrt{3}}{4} -\frac{3}{4} \left(3 \sqrt{3}-2
\pi \right) \right)=7 \pi -9 \sqrt{3}\\
\]

Answer 3

The area in question is
\[
6\pi - 7 \pi + 9 \sqrt 3=9 \sqrt{3}-\pi
\]