Find the area of the region

Question

Find the area of the region that lies inside the curve r = 1 + cos(2theta) but outside the curve

r = 1 + sin(theta)

Answer 1

Let r₁= 1+ cos (2θ) and r₂= 1+sinθ

First you need to find the points of intersection by setting and solving the equation r₁= r_{2 .}

Sometimes the solution of this equation does not reveal all the points of intersection, but in our case works.

Thats why the advice of Mark M. is crucial.

Back to solving the equation r₁= r₂

This implies 1+cos2θ = 1+ sinθ

cos2θ=sinθ

cos2θ= cos(π/2 −θ)

1. (2θ)=2kπ+ π/2 −θ
2. (2θ)=2kπ− π/2 +θ

Solving the above equations you will get θ = π/6 θ= 5π/6 θ= 3π/2 from the first equation and θ

θ=−π/2 and θ = 3π/2.

Because of the symmetry of the graph Α= 4( 1/2)∫ [r₁²-r₂²]dθ from θ=0 to θ= π/6

Please draw a graph