求一道二重积分的计算求∫∫(x²+y²)dxdy,其中区域D为:(x-1)²+y²

求一道二重积分的计算求∫∫(x²+y²)dxdy,其中区域D为:(x-1)²+y²
求一道二重积分的计算
求∫∫(x²+y²)dxdy,其中区域D为:(x-1)²+y²<=1

求一道二重积分的计算求∫∫(x²+y²)dxdy,其中区域D为:(x-1)²+y²
用极坐标变换：x=rcosa,y=rsina,对应的积分区域为(rcosa-1)^2+r^2sin^2a

令x=pcosθ，y=psinθ
x²-2x+y²=0
p²=2pcosθ
p=2cosθ
上半圆为：
{0<=p<=2cosθ
{0<=θ<=π/2
原式=2∫∫p²pdpdθ
=2∫(0,π/2)dθ∫(0,2cosθ)p³dp
=2∫(0,π/2)1/4p^4|(0,2c...

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令x=pcosθ，y=psinθ
x²-2x+y²=0
p²=2pcosθ
p=2cosθ
上半圆为：
{0<=p<=2cosθ
{0<=θ<=π/2
原式=2∫∫p²pdpdθ
=2∫(0,π/2)dθ∫(0,2cosθ)p³dp
=2∫(0,π/2)1/4p^4|(0,2cosθ)dθ
=1/2∫（0，π/2）16cos^4θdθ
=8∫(0,π/2)cos^4θdθ
=8×3/4×1/2×π/2
=(3/2)π

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