Question

Question #e8898

Answer 1

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]}