Hello, rcmango!
Find the area of the region that is inside the circle: .r = 6·cosθ
but outside the cardoid: .r = 2 + 2·cosθ Code:

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Find the intersections of the two curves.
. . 6·cosθ .= .2 + 2·cosθ . → . 4·cosθ .= .2
. . cosθ .= .1/2 . → . θ .= .±π/3
Due to the symmetry, we can integrate from 0 to π/3 and multiply by 2.
A . = . 2 x (1/2) ∫ [(6·cosθ)²  (2 + 2·cosθ)²] dθ
Can you finish it now?