Area is 404.5
Explanation:
Given:
r = 15 cos θ
r = 7 + cos θ
Area of the region, A = ?
When the curves intersect, then the r values are equal.
So,
15 cos θ = 7 + cos θ
15 cos θ - cos θ = 7
14 cos θ = 7
cos θ = [tex]\frac{7}{14}[/tex]
cos θ = [tex]\frac{1}{2}[/tex]
θ = 60°
To find the value of r from 15 cos θ:
r = 15 cos (60°)
r = [tex]15 X\frac{1}{2}[/tex]
r = 7.5
Below is the figure attached for your reference.
Because of symmetry, the shaded area is
[tex]A = 2\int\limits^1_7 {r} \, dr \int\limits^\pi _0 {[14 cos(theta) - 7]} \, d(theta) \\\\A = 2\int\limits^1_7 {r} \, dr [14 sin (theta) - 7(theta)]^\pi ^/^3_0 \\\\A = 2[14 sin(\pi /3) - 7(\pi /3)][\frac{r^2}{2}]^1^5_7_._5\\ \\A = 4.794 X 89.375\\\\A = 404. 49[/tex]
The limit is 7.5 to 15
Therefore, the area of the region is 404.5
