Question

用极坐标计算二重积分：∫∫e∧((x∧2)+(y∧2))dxdy　,其中积分区域D={(x,y)|(x∧2)+(y∧2)<=4}

麻烦各位大侠了

Answer 1

x=rcosa，y=rsina，0<=a<=2pi，0<=r<=2，
Jacobian行列式为r，原积分
=积分（从0到2pi）da 积分（从0到2）e^(r^2)rdr
=2pi* e^(r^2)/2|上限2下限0
=pi*(e^4--1)。

Answer 2

x=ρcosθ
y=ρsinθ
∫∫e^(x²+y²)dxdy
=∫(0→2π)dθ∫(0→2)e^(ρ²)ρdρ
=π∫(0→2)e^(ρ²)dρ²
=πe^(ρ²)|(0→2)
=π(e⁴-1)