find the area of the region inside the limacon r= 8 + 4sin(theta)

Question

HElp!! if you can show steps, i'd be greatful

Answer 1

for polar coordinates, dA = 1/2 r^2 dθ since each tiny wedge of area can be considered a sector of a circle of radius r and angle dθ.

So, since

r = 8+4sinθ,

a = ∫[0,2π] 1/2 (8+4sinθ)^2 dθ
now just expand the quadratic and evaluate each term, and you wind up with
a = 72π

Answer 2

Find the area of the specified region.

Inside the limacon r = 8 + 2 sin θ

Answer 3

66 pi