Question

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

Answer 1

`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..

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