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

设a>0，x=a(cosθ)^3，y=a(sinθ)^3

θ=0时，x=a，y=0；θ=π/2时，x=0，y=a

星形线在第一象限的面积

=∫(θ:π/2→0)ydx

=-∫(θ:0→π/2)ydx

=-∫(0→π/2)a(sinθ)^3*3*a(cosθ)^2*(-sinθ)dθ

=3a^2∫(0→π/2)(sinθ)^4*(cosθ)^2*dθ

=3a^2∫(0→π/2)(sinθ)^4*[1-(sinθ)^2]dθ

=3a^2*[∫(0→π/2)(sinθ)^4*dθ-∫(0→π/2)(sinθ)^6*dθ]

=3a^2*(3/4*1/2*π/2-5/6*3/4*1/2*π/2)

=3πa^2/32

所求图形的面积=a^2-3πa^2/32=(1-3π/32)a^2