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

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Answer 1 · 2016-09-29T02:43:58

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

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..

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