(856,#13) The region inside the circle r=3 cos(theta) and outside the cardiod r = 1 + cos(theta).

The two curves below intersect when 3 cos(theta) = 1 + cos(theta). That is, when 2 cos(theta) = 1, which occurs when theta = Pi/3, or theta =-Pi/3.

plot([3*cos(t), 1+cos(t)], t = 0..2*Pi, coords = polar, color = [red, blue]);

Thus the limits of integration are theta = -Pi/3..Pi/3 and r =1+ cos(theta) .. 3 cos(theta), and the value of theintegral is Pi:

f := (x,y) ->1:
c := t -> 1+cos(t): d := t ->3*cos(t):
Integrate_rdrdt(f, -Pi/3,Pi/3,c,d);

echo:f(r*cos(t), r*sin(t))*r = r c(t) = 1+cos(t) d(t) = 3*cos(t)

int(f(r*cos(t), r*sin(t))*r, r ))= 1/2*r^2

int(f(r*cos(t), r*sin(t))*r, r = c(t)..d(t))= 4*cos(t)^2-1/2-cos(t)

int(int(f(r*cos(t), r*sin(t))*r, r = c(t)..d(t)),t)= 2*cos(t)*sin(t)+3/2*t-sin(t)

int(int(f(r*cos(t), r*sin(t))*r, r = c(t) ..d(t)), t = a..b) = Pi

by:sh