Question

高等数学，多重积分的应用，在线等，重谢！！！！

Answer 1

D: x^2+y^2=ax, 即 r=acost, 0≤r≤acost, -π/2≤t≤π/2.
曲面 z=±√(a^2-x^2-y^2), 则
dS=√[1+(z'<x>)^2+(z'<y>)^2]dxdy=adxdy/√(a^2-x^2-y^2)
=ardrdt/√(a^2-r^2),
由对称性，曲面有相同的2块，得
S = 2∫∫<D>√[1+(z'<x>)^2+(z'<y>)^2]dxdy
= 2∫∫<D>adxdy/√(a^2-x^2-y^2)
因D关于x轴对称，积分函数是y的偶函数，则
S = 4∫<0,π/2>dt∫<0,acost>ardr/√(a^2-r^2)
= 2a∫<0,π/2>dt∫<0,acost>[-d(a^2-r^2)]/√(a^2-r^2)
= 2a∫<0,π/2>dt[-2√(a^2-r^2)]<0,acost>
= 4a^2∫<0,π/2>（1-sint）dt
=4a^2[t+cost]<0,π/2> = 4a^2(π/2-1) = 2(π-2)a^2.