r=a(1+sinθ)
r=asinθ
0≤θ≤π
The area of the circle
A1=21∫0π(asinθ)2dθ=4a2∫0π(1−cos(2θ))dθ
=4a2[θ−21sin(2θ)]π0=4πa2(units2)
The area of the part of the cardioid
A2=21∫0π(a(1+sinθ))2dθ
=2a2∫0π(1+2sinθ+21−21cos(2θ))dθ
=2a2[23θ−2cosθ−41sin(2θ)]π0
=a2(43π+2)(units2)
Area=A2−A1=43πa2+2a2−4πa2
=(2π+2)a2 (units2)
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