1.
Change variable to polar coordinate,i.e. x = rcosθ, y = rsinθ, then Ω: 0 ≤ r ≤ 1, 0≤ θ ≤ π.
∫ ∫
Ω
ln(x2+ y2+ 1) dA
=
∫ π
0
∫ 1
0
ln(r2+ 1)r dr dθ
=π 2
∫ 2 1
lnu du (make change of variable u = r2+ 1).
=π
2[ulnu|21−
∫ 2
1
u1
udu] (use integral by part.)
=π
2(2ln2− 1)
1
2.
Let new variable x = u− 2v, y = 3u + v, then Ω is a region bounded by y − x = 0, y = 0, x = 0.
∫ ∫
Ω
=√
u− 2v +√
3u− v du dv
=
∫ 1
0
∫ x
0
√x +√ y|
ux uy vx vy
|dy dx
=1 7
∫ 1
0
∫ x
0
√x +√
y dy dx
=1 7
∫ 1 0
5 3x32 dx
= 2 21
2
