急求星形线x 2 3 y 2 3 a 2 3 与x a及y a围成的图形的面积 谢谢

化成参数方程,

x=a(cost)^3,

y=a(sint)^3,

图形星形线第一象限和正方形之间所围成图形,

S=∫ [0,a] ydx=4∫ [π/2,0] a(sint)^3d[a (cost)^3]

=a^2∫ [π/2,0] (sint)^3 *[3(cost)^2*(-sint)]dt

=(-3a^2)∫ [π/2,0](sint)^4(cost)^2dt

=(-3/8)∫ [π/2,0] [1-2cos2t+(cos2t)^2](1+cos2t)dt

=(-3/8)a^2 [π/2,0][t-t/2-sin4t/8+(1/2)sin2t-(sin2t)^3/3]

=(-3a^2/8)[0-(π/4-0+0-0]

=3πa^2/32,

第一象限星形线外是正方形,面积为a^2,

∴二者所围成面积为:a^2-3πa^2/32.