How do you find the area ( if any ) common to the four cardioids r=1+-cos theta and r = 1+-sin theta?

1 Answer
Sep 29, 2016

3/2pi-4 sqrt 2 +1=0.055535 areal units.

Explanation:

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

(1-1/sqrt 2, pi/4) in Q_1

(1-1/sqrt 2, 3/4pi) in Q_2

(1-1/sqrt 2, 5/4 pi) in Q_3

(1-1/sqrt 2, 7/4pi) in Q_4

Now, one such part is the area in Q_3 that is

symmetrical about theta = 5/4pi and bounded by

r = 1+cos theta, between (0, pi) and (1-1/sqrt 2, 5/4pi) and

r = 1+sin theta, between (0, 3/2pi) and (1-1/sqrt 2, 5/4pi)

This area = 2 int int r dr d theta,

r from 0 to (1+cos theta) and theta from pi t.o 5/4pi.

After integration with respect to r, this becomes

int (1+cos theta)^2 d theta, between the limits pi and 5/4pi.

= int (1+2 cos theta + cos^2theta) d theta, between the limits

= int (1+2 cos theta + (1+ cos 2theta)/2) d theta, between the limits

= [3/2 theta + 2 sin theta +1/4sin 2 theta], between pi and 5/4pi.

= 3/8pi-2/sqrt 2+1/4

=(3pi-8 sqrt 2 + 2)/8

The total common area is 4 X this area

= 3/2pi-4 sqrt 2 +1=0.055535 areal units

I welcome a graphical depiction for my answer..

.

,