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

It is ##pi/4##

Find the intersection points of the curves hence we have that

##3cosθ=1+cosθ=>cosθ=1/2=>θ=+-pi/3##

The saded area is

(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 ##pi/8##

The total amount is ##2pi/8=pi/4##

A graph for the curves is