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]02π=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).)