How do you find the area ( if any ) common to the four cardioids #r=1+-cos theta and r = 1+-sin theta#?
The lines of symmetry (axes ) of the cardioids are the positive and
negative axes of coordinates.
The cardioids are equal in size and symmetrically placed, with
respect to the pole r=0 that is a common point of imtersection for all.
The common area comprises four equal parts.
The other terminal points, in these parts, are
Now, one such part is the area in
This area =
r from 0 to
After integration with respect to r, this becomes
The total common area is 4 X this area
I welcome a graphical depiction for my answer..