X=acos^3t,y=asin^3t 所 围成的平面图形的面积

题目

X=acos^3t,y=asin^3t 所 围成的平面图形的面积

答案

x=acos^3t,y=asin^3t是星形线,它的面积为
∫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