二重积分∫∫D√(x^2+y^2)dσ,其中D是由抛物线x^2+y^2=1,x^2+y^2=4围成?

2025-05-19 22:22:18
推荐回答(1个)
回答1:

换成极坐标x=ρcosθ,y=ρsinθ
积分区域为1≤x²+y²≤4,x≥0,y≥0
即1≤ρ²≤4,cosθ≥0,sinθ≥0
则ρ∈[1,2],θ∈[0,π/2]
∫∫D ∨(x²+y²)dσ
=∫(0,π/2) dθ∫(1,2) ρ²dρ
=π/6 ρ³|(1,2)
=7π/6