Question

利用极坐标计算二重积分？

如图第三小题！

Answer 1

转化为极坐标是 ∫<0,π/2>dθ ∫<0,1/(cosθ+sinθ)>r*e^[sinθ/(cosθ+sinθ)]dr

Answer 2

∫∫e^(y/(x+y))dxdy
使用极坐标，
x=rcosθ，
y=rsinθ
r∈(0,1/(sinθ+cosθ)),
θ∈(0,π/2)
=∫(0,π/2)dθ∫(0,1/(sinθ+cosθ)）e^(sinθ/(sinθ+cosθ))rdr
=(1/2)∫（0,π/2）e^(sinθ/(sinθ+cosθ))dθ/(sinθ+cosθ)²
=(1/2)∫（0,π/2）e^(sinθ/(sinθ+cosθ))d(sinθ/(sinθ+cosθ))
=(1/2)e^(sinθ/(sinθ+cosθ))│（0,π/2）
=(1/2)(e^(1/(1+0))-e^(0/(0+1)))
=(e-1)/2。