Question

求积分I= ∫ ∫根号（x^2+y^2）dxdy积分区域是D,其中D由y=x与y=x^4围成.用极坐标的方法.

Answer 1

y=x => θ=π/4

y=x^4 => rsinθ=(rcosθ)^4 => r^3=sinθ/(cosθ)^4 => r=[sinθ/(cosθ)^4]^(1/3)

I=∫[0->π/4]∫[0->[sinθ/(cosθ)^4]^(1/3)] r√r² drdθ=∫[0->π/4] sinθ/3(cosθ)^4 dθ

=-∫[0->π/4] 1/3(cosθ)^4 dcosθ= 1/9cos³θ | [0->π/4] =4/(9√2)