Find the area of the region that lies inside the first curve and outside the second curve. r = 15 cos(θ), r = 7 + cos(θ)

The first step is to determine the bounds that define the area between the two given curves.

Set 15cos(θ) = 7 + cos(θ) to obtain the value of θ where the curves intersect.
15cos(θ) = 7 + cos(θ)
14cos(θ) = 7
cos(θ) = 1/2  => θ = +60° or -60°.

From r = 15cos(θ), obtain
r = 15*(1/2) = 7.5

The shaded area of the curve between the two curves is shown in the figure below.

Because of symmetry, the shaded area is

=
= 
= (4.794)*(89.375)
= 404.4915

404.5 (nearest tenth)