1

0

! −1−

√1−y2

−

√4−y2 f(x, y)dxdy + ! 1

0

! 0

−1+√1−y2 f(x, y)dxdy + ! 2

1

! 0

−

√4−y2 f(x, y)dxdy

Ωn = {(x, y, z) x ≥ 0 , y ≥ 0 , x2 + y2 ≤ n2 , 0 ≤ z ≤ n}

, !

Ωn sen (x2 + y2) dV = ! π/2

0

! n

0

! n

0 sen (r2)rdzdrdθ = π

4n(1 − cos (n2)

5π/12

23π/160

Ωn = {(x, y, z) ∈ IR3 , 1

n2 ≤ x2 + y2 ≤ 1 , −1 ≤ z ≤ 1}, onde n ≥ 2.

e !

Ωn |Ln(

"x2 + y2)| dV ≤ 4

limn→∞ # 2π

0

# 1

1/n

# 1

−1

rLn(r)dzdrdθ = −π

θ.

!

D x dA = −π

a (D1)n = {(x, y) ∈ IR2 , 1 ≤ x2 + y2 ≤ n2}; n ≥ 2. Tem-se !

(D1)n |f|dA ≤

! 2π

0

! n

1 r−3/2drdθ ≤ M

(D2)n = {(x, y) ∈ IR2 , 1

n2 ≤ x2 + y2 ≤ 1}. !

(D2)n |f|dA ≤ ! 2π

0

! 1

1/n r−1/2drdθ ≤ M˜

D3 = D1 ∪ D2

T −1 dada por u = xy, v = y

x . u = 1, u = 2, v = 1/2 e v = 2 e |JT −1 | = 2y

x , logo o |JT | = x

2y !

D(x2 + y2) dA = ! 2

1

! 2

1/2(u

2 + u

2v2 )dvdu = 9

4 . D = {(x, y) , 1 ≤ y ≤ e , 0 ≤ x ≤ Ln(y)}.

Dn = {(x, y) , 1 + 1

n ≤ y ≤

e , 0 ≤ x ≤ Ln(y)}. = !

Dn |f|dA ≤ 27 = limn→∞ !

Dn f dA = limn→∞ 1

2 (ee − e1+ 1

n ) = ee−e

2 .