Question #e8898

1 Answer
Feb 2, 2017

Answer:

The common points are #(3/2, -pi/3) and (3/2, pi/3)#.
Area inside the circle and outside the cardioid is #9(sqrt3-pi/3)#
= 6.16368 areal units, nearly. See graph..

Explanation:

At the common points of the circle

#r = 3costheta# and the cardioid

#r = 3(1-costheta)#,

#r =3costheta=3(1-costheta)#, giving #costheta=1/2#

In #(-pi, pi)#, the angles #alpha and beta# as solutions of this

equation are #+-pi/3#.

The area A inside the circle and outside the cardioid is symmetrical

about #theta = 0#. So,

A = #2 (1/2int r^2 d theta#, for the circle

#-1/2 int r^2 d theta#. for the cardioid),

with #theta# from 0 to #pi/3#

#=9int(cos^2theta-(1-costheta)^2) d theta#, for the limits

#=9 int(2costheta-1) d theta#, for the limits

#=9[2sintheta-theta]#, between #0 and pi/3#

#=9(sqrt3-pi/3)#

graph{(x^2+y^2-3x)(x^2+y^2-3sqrt(x^2+y^2)+3x)=0x^2 [-10, 10, -5, 5]}