计ç®äºé积åâ«â«x²e^(-y²)dxdyï¼å
¶ä¸Dæ¯ä»¥(0ã0)ã(1ã1)å(0,1)为顶ç¹çä¸è§å½¢åºåã
解ï¼ãDãâ«â«x²e^(-y²)dxdy=ã0ï¼1ãâ«e^(-y²)dyã0ï¼yãâ«x²dx
=ã0ï¼1ãâ«{e^(-y²)[(1/3)x³]ã0ï¼yã}dy
=ã0ï¼1ã(1/3)â«[y³e^(-y²)]dy
=ã0ï¼1ã(-1/6)â«y²d[e^(-y²)]
=ã0ï¼1ã(-1/6)[y²e^(-y²)+â«e^(-y²)d(-y²)]
=(-1/6)[y²e^(-y²)+e^(-y²)]ã0ï¼1ã
=(-1/6)[(1/e)+(1/e)-1]
=(1/6)-1/(3e).
解ï¼ãDãâ«â«x²e^(-y²)dxdy=ã0ï¼1ãâ«e^(-y²)dyã0ï¼yãâ«x²dx
=ã0ï¼1ãâ«{e^(-y²)[(1/3)x³]ã0ï¼yã}dy
=ã0ï¼1ã(1/3)â«[y³e^(-y²)]dy
=ã0ï¼1ã(-1/6)â«y²d[e^(-y²)]
=ã0ï¼1ã(-1/6)[y²e^(-y²)+â«e^(-y²)d(-y²)]
=(-1/6)[y²e^(-y²)+e^(-y²)]ã0ï¼1ã
=(-1/6)[(1/e)+(1/e)-1]
=(1/6)-1/(3e).
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