4. 化为极坐标后 D: 1≤r≤2, 0≤t≤π/4.
∫∫dxdx/(1+x^2+y^2) = ∫dt∫ rdr/(1+r^2)
=∫dt∫ (1/2)d(1+r^2)/(1+r^2)
= (1/2)∫dt[ln(1+r^2)]
= (1/2)∫dt[ln(1+r^2)] = π(ln5-ln2)/8.
5. f(x,y)=3axy-x^3-y^3, f'=3ay-3x^2, f'=3ax-3y^2,
得驻点 (0,0),(a,a).
f''=-6x, f''=3a, f''=-6y
对于点(0,0), A=C=0， B=3a， AC-B^2=-9a^2