Question

二重积分的取值范围为,其中d是圆形闭区域

求二重积分∫∫√x2+y2dδ,其中D是圆环形闭区间{(x,y)|a2<=x2+y2<=b2}
∫∫√x2+y2 dδ,其中D是圆环形闭区间{(x,y)|a2<=x2+y2<=b2}

Answer 1

换为极坐标,环形区域为a≤r≤b,0≤θ≤2π
√(x²+y²)=r,dσ=dxdy=rdrdθ
∫∫√(x²+y²)dσ
=∫∫r*rdrdθ
=∫dθ*∫r²dr
=2π*[r³/3]
=2π*(b³-a³)/3