曲线y=√x变为x=y²
∫∫e^(x/y)dxdy
=∫(0,1)∫(y²,y) e^(x/y)dxdy
先积里面的积分,将y看作常量
∫(y²,y) e^(x/y)dx
=[y*e^(x/y)] (y²,y)
=[y*e^(y/y)]-[y*e^(y²/y)]
=ye-ye^y
再积外面的
∫(0,1) (ye-ye^y)dy
先积前面的
∫eydy=ey²/2
后面的要用到分部积分法
∫ye^ydy=ye^y-∫e^ydy=ye^y-e^y=(y-1)e^y
所以∫(0,1) (ye-ye^y)dy
=[ey²/2-(y-1)e^y] (0,1)
=[e/2-(1-1)*e]-[0-(0-1)*1]
=(e/2)-1
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(0,1)表示下限是0,上限是1