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
[tex]A=2 \int\limits^{15}_{7.5} {rdr} \, \int\limits^{ \pi /3}_ {0} \, [14cos( \theta) -7]d \theta [/tex]
= [tex]=2 \int\limits^{15}_{7.5} {r} \, dr[14sin \theta-7 \theta ]^{ \pi /3}_{0} [/tex]
= [tex]2[14sin( \pi /3) - 7( \pi /3)][ \frac{r^{2}}{2} ]^{15}_{7.5}[/tex]
= (4.794)*(89.375)
= 404.4915
Answer: 404.5 (nearest tenth)
