二重积分：计算∫∫e^(y/(x+y))dxdy,其中D:x+y≤1,x≥0,y≥0.

Question

如题所述

Answer 1

简单计算一下，答案如图所示

Answer 2

解：∫∫<D>e^(y/(x+y))dxdy=∫<0,π/2>dθ∫<0,1/(sinθ+cosθ)>e^(sinθ/(sinθ+cosθ))rdr (做极坐标变换)
=(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。