Integrales Dobles

INTEGRALES DOBLES
1) Calcula las siguientes integrales dobles:
x2 y d(x, y), M = [0, 1] × [0, 1]

(1)
M

(2)
M

(3)

x2
d(x, y), M = [0, 1] × [0, 1]
1 + y2
(log x)y d(x, y), M = [1, e] × [1, e]

M(4)

(log x)y d(x, y), M = (0, 1] × [0, 1]
M

x3 y 3 d(x, y), M = [0, 1] × [0, 1]

(5)
M

x log(xy) d(x, y), M = [2, 3] × [1, 2]

(6)
M

2

2

(ex − ey ) d(x, y), M = [a, b] × [a, b]

(7)
M

(8)
M

1d(x, y), M = [3, 4] × [1, 2]
(x + y)2
y 2 sen(xy) d(x, y), M = [0, 2π] × [0, 1]

(9)
M

x
d(x, y), M = [−π/2 , π/2] × [1, 2].
y
1
π
π
d(x, y), M = [0, ] × [0, ]
2
(1 + x)
2
2
1
3π
d(x, y), M = [0, 10]× [0,
]
2
(1 + x)
2

ey sen

(10)
M

(11)
M

(12)
M

2) Sea f : M −→ R donde M = [−1, 1] × [−1, 1] tal que f (x, −y) = −f (x, y) para todo
(x, y) ∈ M . Probar que
f (x, y) d(x, y) = 0.
M

Calcula

Msen(xy) d(x, y).

3) Calcula las siguientes integrales dobles:
(1)

M

√x
y+1

(2)

M

x d(x, y), M = {(x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ ex }

(3)

x2
M y2

d(x, y), M = {(x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ x}

d(x,y), M = {(x, y) : 1 ≤ x ≤ 2, x−1 ≤ y ≤ x}
1

(4)

M

ex/y d(x, y), M = {(x, y) : 1/2 ≤ y ≤ 1, 0 ≤ x ≤ y 2 }

(5)

M

xy 2 d(x, y), M = {(x, y) : |y| ≤ x ≤ 1}

(6)

M

xy d(x, y), M es la regi´ondelimitada por las curvas y = x e y = x2 .

(7)
(8)

xy − y 2 d(x, y), donde M es el tri´angulo de v´ertices (0, 0), (1, 1)
y (10, 1).
M

M

x d(x, y), M es el subconjunto del semiplano x ≥ 0 cuyafrontera son las curvas
x2
y2
−
=1
a2
b2

(9)

M

x2
y2
+
= 1.
4a2
9b2

y

|x2 − y + 1| d(x, y), M = {(x, y) : y ≥ 0, x2 + y 2 ≤ 4}

(10)

M

|x + y| d(x, y), M = {(x, y) : |x| ≤ 1, |y| ≤ 1}

(11)

M

xy dxdy con M = {(x, y) ∈ R : 0 ≤ x ≤ 2, 0 ≤ y ≤ x/2}

(12)

M

x2 y dx dy con M = {(x, y) ∈ R : x ≥ 0, y ≥ 0, x + y ≤ 1}

(13)

M

y 3 dx dy con M = {(x, y) ∈ R : x ≤ a, y 2 ≤ 2px}, a > 0, p > 0

(14)M

y/(1 + x3 ) dx dy con M = {(x, y) ∈ R : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, x2 + y 2 ≥ 1}

(15)

M

ex+y dx dy con M el tri´angulo de v´ertices (1, 0), (0, 1) y (0, −1).

2

2

2

2

(16)

xy dx dy con M el...