二重积分计算题，不明白最后一步是如何得出的，求推导过程。 求帮帮忙解一道二重积分计算题
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\u6240\u4ee5\u5176\u503c\u4e3a\u79ef\u5206\u533a\u57dfDz\u7684\u9762\u79ef\u3002
Dz\u662f\uff0c\u5bf9\u4e8e-c\u5230c\u4e0a\u53d6\u7684z\uff0c\u7528\u5e73\u9762z=z\u622a\u692d\u7403\u6240\u5f97\u7684\u692d\u5706(\u89c1\u56fe\u4e2d\u9634\u5f71)
\u6240\u4ee5\uff0c\u8be5\u4e8c\u91cd\u79ef\u5206\u662f\u692d\u5706\u7684\u9762\u79ef\u3002
\u692d\u5706\u7684\u65b9\u7a0b\u662f xx/aa + yy/bb =1- zz/cc\uff0c
\u8bb0\u4e0a\u5f0f\u53f3\u7aef\u4e3a\u2605\uff0c\u5219\u692d\u5706\u7684\u4e24\u8f74\u4e3aa\u221a\u2605\u53cab\u221a\u2605\uff0c
\u4e8e\u662f\u5f97\u5230\u692d\u5706Dz\u7684\u9762\u79ef=\u03c0ab\u2605\u3002
\u8be6\u7ec6\u8fc7\u7a0b\u5982\u56feet\u2026\u2026\u5e0c\u671b\u80fd\u5e2e\u5230\u4f60\u89e3\u51b3\u95ee\u9898
令：F = (-∞,+∞)∫e^(-x²)dx
同样 F= (-∞,+∞)∫e^(-y²)dy
由于x,y是互不相关的的积分变量,因此：
F² = (-∞,+∞)∫e^(-x²)dx * (-∞,+∞)∫e^(-y²)dy
= [D]∫∫e^(-x²)*dx * e^(-y²)*dy
= [D]∫∫e^[-(x²+y²)]*dx *dy
式中积分域D = {(x,y)|x ∈(-∞,+∞),y∈(-∞,+∞)}
对x,y进行极坐标变换,则:
x²+y² = ρ²；dxdy = ρ*dρ*dθ
F² = [D]∫∫e^[-(x²+y²)]*dx *dy
= [0,+∞)[0,2π]∫∫e^(-ρ²) ρ*dρ*dθ
= [0,2π]∫dθ *(0,+∞)∫e^(-ρ²) ρ*dρ
= 2π* 1/2*[0,+∞)*∫e^(-ρ²) *dρ²
= π
因此 F = (-∞,+∞)∫e^(-x²)dx = √π
一半就是你要的那个
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