计算二重积分：∫∫（D）ydxdy,其中D：x^2+y^2≤2x,y≥0

Question

计算二重积分：∫∫（D）ydxdy,其中D：x^2+y^2≤2x,y≥0

Answer 1

变成极坐标啊
令x=pcosa
y=psina
代入
x^2+y^2≤2x
p^2≤2pcosa
p≤2cosa
由于y≥0,所以0≤a≤π
∫∫（D）ydxdy
=∫[0,π]∫[0,2cosa] psina*pdpda
=∫[0,π]sina*p^3/3[0,2cosa]da
=8/3∫[0,π]sina*(cosa)^3da
=-8/3∫[0,π](cosa)^3dcosa
=-2/3(cosa)^4[0,π]
=4/3