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积分区域D内的点(x,y)
x+y∈[0,1]
x+y≥(x+y)²
∫∫(x+y)dxdy>∫∫(x+y)²dxdy
选择A
y=-x与y=x+1交点为(-1/2,1/2)
∫∫D (x+y)dxdy
=∫(-1/2,0) dx∫(-x,x+1) (x+y)dy
=∫(-1/2,0) (2x²+2x+1/2)dx
=(2/3 x³+x²+1/2 x)|(-1/2,0)
=1/12
∫∫D (x+y)²dxdy
=∫(-1/2,0)dx∫(-x,x+1) (x²+2xy+y²)dy
=∫(-1/2,0) (8/3 x³+4x²+2x+1/3)dx
=(2/3 x^4+4/3 x³+x²+1/3 x)|(-1/2,0)
=-1/24 +1/6 -1/4+1/6
=1/24
x+y∈[0,1]
x+y≥(x+y)²
∫∫(x+y)dxdy>∫∫(x+y)²dxdy
选择A
y=-x与y=x+1交点为(-1/2,1/2)
∫∫D (x+y)dxdy
=∫(-1/2,0) dx∫(-x,x+1) (x+y)dy
=∫(-1/2,0) (2x²+2x+1/2)dx
=(2/3 x³+x²+1/2 x)|(-1/2,0)
=1/12
∫∫D (x+y)²dxdy
=∫(-1/2,0)dx∫(-x,x+1) (x²+2xy+y²)dy
=∫(-1/2,0) (8/3 x³+4x²+2x+1/3)dx
=(2/3 x^4+4/3 x³+x²+1/3 x)|(-1/2,0)
=-1/24 +1/6 -1/4+1/6
=1/24
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