find the area of the specified region

Question

inside the limacon r= 7+6sin

Answer 1

r=7+sinθ. The area inside the limacon=½∫r²dθ for θ∈[0,2π].

Area=½∫(49+14sinθ+sin²θ)dθ for θ∈[0,2π].

cos(2θ)=1-2sin²θ, so sin²θ=½(1-cos(2θ)).

Area=½∫(49+14sinθ+½(1-cos(2θ)))dθ for θ∈[0,2π]=

½∫(49½+14sinθ-½cos(2θ))dθ for θ∈[0,2π]=

½[99π/2-14cosθ-sin(2θ)/4]₀^2π=½(99π-14+14)=99π/2 sq units (about 155.51).

(This limacon is almost a perfect circle with radius 7. The area of such a circle is 49π=(153.94).)