# How do you find the region inside cardioid r=1+cos(theta) and outside the circle r=3cos(theta)?

It is $\frac{\pi}{4}$

#### Explanation:

Find the intersection points of the curves hence we have that

3cosθ=1+cosθ=>cosθ=1/2=>θ=+-pi/3

(cardiod area from pi/3 to pi)-(cricle area from pi/3 to pi/2)

The cardiod area is

int_(pi/3)^(pi) 1/2*(1+cosθ)^2dθ=pi/2-9/6*sqrt3

and the circle area is

int_(pi/3)^(pi/2) 1/2*(3*cosθ)^2dθ=(3pi/8)-9/16*sqrt3

Hence the shaded area is $\frac{\pi}{8}$

The total amount is $2 \frac{\pi}{8} = \frac{\pi}{4}$

A graph for the curves is