计算二重积分.∫∫(x+y)/(x^2+y^2)dσ,D:x^2+y^2≤1及x+y≥1所确定.呵呵,
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运用极坐标代换
x=rcosa,y=rsina,a[0,π/2],r[0,1],dxdy=rdrda
x+y=rcosa+rsina=√2rsin(a+π/4)
∫∫(x+y)/(x^2+y^2)dσ
=∫[0,π/2]∫[0,1]√2rsin(a+π/4)/r^2*rdrda
=∫[0,π/2]√2sin(a+π/4)da
=-cos(a+π/4)[0,π/2]
=√2/2+1
x=rcosa,y=rsina,a[0,π/2],r[0,1],dxdy=rdrda
x+y=rcosa+rsina=√2rsin(a+π/4)
∫∫(x+y)/(x^2+y^2)dσ
=∫[0,π/2]∫[0,1]√2rsin(a+π/4)/r^2*rdrda
=∫[0,π/2]√2sin(a+π/4)da
=-cos(a+π/4)[0,π/2]
=√2/2+1
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