A=210∫2πr2dθ=2a20∫2π(1+sinθ)2dθ=2a20∫2π(1+2sinθ+sin2θ)dθ==2a20∫2π(1+2sinθ+21−cos2θ)dθ=2a2(θ−2cosθ+θ/2−4sin2θ)∣∣02π=2a2(2π−2+π−0)−2a2(0−2+0−0)=23a2π
R∬xdA=0∫2π0∫a(1+sinθ)x⋅r dr dθ=0∫2π0∫a(1+sinθ)r2cosθ dr dθ==0∫2π31a3(1+sinθ)3cosθ dθ=12a3(1+sinθ)4∣∣02π=0
R∬ydA=0∫2π0∫a(1+sinθ)y⋅r dr dθ=0∫2π0∫a(1+sinθ)r2sinθ dr dθ==0∫2π31a3(1+sinθ)3sinθ dθ=3a30∫2π(sinθ+3sin2θ+3sin3θ+sin4θ)dθ==3a30∫2π(3sin2θ+sin4θ)dθ=3a30∫2π(3sin2θ+(1−cos2θ)sin2θ)dθ==3a30∫2π(4sin2θ−sin2θcos2θ)dθ=3a30∫2π(2(1−cos2θ)−41sin22θ)dθ==3a30∫2π(2−2cos2θ−81+81cos4θ)dθ=3a3(815θ−sin2θ+321sin4θ)∣∣02π=1215a3π
x=A1R∬xdA=0
y=A1R∬ydA=1215a3π:23a2π=65a
Answers: centroid is (0, 65a) .
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