x=acos^3t,y=asin^3t是星形線，它的面積為

∫ydx=4*∫asin^3t(acos^3t)"dt,t:π/2→0

=-3*a^2∫sin^4t*cos^2tdt

=-3a^2∫(sin^4t-sin^6t)dt

=3/8*πa^2

只要計算第一象限部分的長度,再乘以4即可首先,弧微分ds＝√[(dx)^2＋(dy)^2]＝√[(x")^2＋(y")^2]dt＝3a|sintcost|dt, x"、y"表示求導其次,弧長s＝4∫(0,π/2) 3a|sintcost|dt＝12a∫(0,π/2) sintcostdt＝6a

用格林公式求星型線 x=acos³t,y=asin³t的面積. S=(1/2)∮xdy-ydx=[0,2π](1/2)∫(3a²cos⁴tsin²t+3a²sin⁴tcos²t)dt =[0,2π](3a²/2)∫(cos²tsin²t(cos²t+sin²t)dt=[0,2π](3a²/2)∫(cos²tsin²t)dt =[0,2π](3a²/2)∫[(1/4)(1+cos2t)(1-cos2t)dt=[0,2π](3a²/2)∫[(1/4)(1-cos²2t)dt =[0,2π](3a²/2)[(1/8)∫dt-(1/32)∫cos4td(4t)] =(3a²/2)[t/8-(1/32)sin4t][0,2π]=(3/8)πa²