星形线x=acos^3t,y=asin^3t(a>0)所围成的面积

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星形线x=acos^3t,y=asin^3t(a>0)所围成的面积

星形线x=acos^3t,y=asin^3t(a>0)所围成的面积
星形线x=acos^3t,y=asin^3t(a>0)所围成的面积

星形线x=acos^3t,y=asin^3t(a>0)所围成的面积
由对称性,
S=4∫(0→a)ydx
=4∫(π/2→0) a(sint)^3 d[a(cost)^3]
=12a^2×∫(0→π/2) (sint)^4×(cost)^2 dt
=12a^2×∫(0→π/2) [(sint)^4-(sint)^6] dt
=12a^2×[3/4×1/2×π/2-5/6×3/4×1/2×π/2]