急求星形线x^(2/3)+y^(2/3)=a^(2/3)与x=a及y=a围成的图...

Question

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

Answer 1

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.