计算二重积分:∫[0,1]dx∫[0,x^½]e^(-y²/2)dy
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计算二重积分:∫[0,1]dx∫[0,x^½]e^(-y²/2)dy
原式=∫dy∫e^(-y²/2)dx (作积分顺序变换)
=∫(1-y²)e^(-y²/2)dy
=∫e^(-y²/2)dy-∫y²e^(-y²/2)dy
=∫e^(-y²/2)dy-{[-ye^(-y²/2)]│+∫e^(-y²/2)dy} (应用分部积分法)
=∫e^(-y²/2)dy-[-e^(-1/2)+∫e^(-y²/2)dy]
=∫e^(-y²/2)dy+e^(-1/2)-∫e^(-y²/2)dy
=e^(-1/2)
=1/√e.
=∫(1-y²)e^(-y²/2)dy
=∫e^(-y²/2)dy-∫y²e^(-y²/2)dy
=∫e^(-y²/2)dy-{[-ye^(-y²/2)]│+∫e^(-y²/2)dy} (应用分部积分法)
=∫e^(-y²/2)dy-[-e^(-1/2)+∫e^(-y²/2)dy]
=∫e^(-y²/2)dy+e^(-1/2)-∫e^(-y²/2)dy
=e^(-1/2)
=1/√e.
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