﹙x²+y²﹚³=x∧4+y∧4所围区域的面积

Question

﹙x²+y²﹚³=x∧4+y∧4所围区域的面积

Answer 1

令x=rcosθ,y=rsinθ
原方程化成极坐标方程r^6=r^4[(cosθ)^4+(sinθ)^4],化简得：r(θ)=√[(cosθ)^4+(sinθ)^4]=√[1-2(cosθsinθ)^2]=√{1-[sin(2θ)]^2/2}
故所围成的区域面积就是在0≤θ≤2π上的积分.
S=(1/2)∫[0,2π] r²(θ)dθ=(1/2)∫[0,2π] {1-[sin(2θ)]^2/2}dθ=∫[0,2π] [3/8+cos(4θ)/8]dθ=3π/4