Oitava Lista de Exerc´ıcios - feg.unesp.br · Oitava Lista de Exerc´ıcios 1. ... xlnydxdy, onde...
Transcript of Oitava Lista de Exerc´ıcios - feg.unesp.br · Oitava Lista de Exerc´ıcios 1. ... xlnydxdy, onde...
Oitava Lista de Exercıcios
1. Seja A o retangulo 1 ≤ x ≤ 2, 0 ≤ y ≤ 1. Calcule∫∫
Af(x, y)dxdy sendo f(x, y) igual a
(a) f(x, y) = ycosxy.
(b) f(x, y) = 1(x+y)2
.
(c) f(x, y) = yexy.
(d) f(x, y) = xy2.
2. Calcule
(a)∫∫
Axlnydxdy, onde A e o retangulo 0 ≤ x ≤ 1, −π
4≤ y ≤ π
4.
(b)∫∫
Axyex
2−y2dxdy, onde A e o retangulo −1 ≤ x ≤ 1, 0 ≤ y ≤ 3.
(c)∫∫
Asen2x1+4y2
dxdy, onde A e o retangulo 0 ≤ x ≤ π2, 0 ≤ y ≤ 1
2.
3. Calcule o volume do conjunto dado
(a) {(x, y, z) ∈ R3 : 0 ≤ x ≤ 2, 1 ≤ y ≤ 2, 0 ≤ z ≤ √
xy}.(b) {(x, y, z) ∈ R
3 : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, x2 + y2 ≤ z ≤ 2}.(c) {(x, y, z) ∈ R
3 : 1 ≤ x ≤ 2, 0 ≤ y ≤ 1, x+ y ≤ z ≤ x+ y + 2}.(d) {(x, y, z) ∈ R
3 : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 1 ≤ z ≤ ex+y}.
4. Calcule Calcule∫∫
Bydxdy, onde B e o conjunto dado
(a) B e o triangulo de vertices (0, 0), (1, 0) e (2, 1).
(b) B e a regiao compreendida entre os graficos de y = x e y = x2, com 0 ≤ x ≤ 2.
(c) B e o paralelogramo de vertices (−1, 0), (0, 0), (1, 1) e (0, 1).
(d) B e o semicırculo x2 + y2 ≤ 4, y ≥ 0.
5. Calcule∫∫
Bf(x, y)dxdy sendo dados
(a) f(x, y) = xy√
x2 + y2 e B o retangulo 0 ≤ x ≤ 1, 0 ≤ y ≤ 1.
(b) f(x, y) = x+ y e B o paralelogramo de vertices (0, 0), (1, 1), (3, 1) e (2, 0).
(c) f(x, y) =1
ln ye B =
{
(x, y) ∈ R2 : 2 ≤ y ≤ 3, 0 ≤ x ≤ 1
y
}
.
(d) f(x, y) = xycosx2 e B = {(x, y) ∈ R2 : 0 ≤ x ≤ 1, x2 ≤ y ≤ 1}.
(e) f(x, y) = x + y e B a regiao compreendida entre os graficos das funcoes y = x ey = ex, com 0 ≤ x ≤ 1.
(f) f(x, y) = y3exy2e B o retangulo 0 ≤ x ≤ 1, 1 ≤ y ≤ 2.
(g) f(x, y) = x5y3 e B = {(x, y) ∈ R2 : y ≥ x2, x2 + y2 ≤ 2}.
(h) f(x, y) = x2 e B o conjunto de todos os (x, y) tais que x ≤ y ≤ −x2 + 2x+ 2.
(i) f(x, y) = 1 e B o conjunto de todos os (x, y) tais que y ≥ x2 e x ≤ y ≤ x+ 2.
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6. Inverta a ordem de integracao
(a)
∫ 1
0
[∫ x
0
f(x, y)dy
]
dx.
(b)
∫ 1
−1
[
∫
√1−x2
−√1−x2
f(x, y)dy
]
dx.
(c)
∫ 1
0
[∫ 1
x2
f(x, y)dy
]
dx.
(d)
∫ 1
0
[
∫
√2x
√x−x2
f(x, y)dy
]
dx.
(e)
∫ π4
0
[∫ cosx
senx
f(x, y)dy
]
dx.
(f)
∫ 2
−1
[
∫ x+73
√
7+5x2
3
f(x, y)dy
]
dx.
(g)
∫ 3
0
[
∫
√3x
x2−2x
f(x, y)dy
]
dx.
7. Calcule o volume do conjunto dado
(a) {(x, y, z) ∈ R3 : x ≥ 0, y ≥ 0, x+ y ≤ 1 e 0 ≤ z ≤ x2 + y2}.
(b) {(x, y, z) ∈ R3 : 0 ≤ y ≤ 1− x2 e 0 ≤ z ≤ 1− x2}.
(c) {(x, y, z) ∈ R3 : x2 + 4y2 ≤ 4 e x+ y ≤ z ≤ x+ y + 1}.
(d) {(x, y, z) ∈ R3 : x ≥ 0, x ≤ y ≤ 1 e 0 ≤ z ≤ ey
2}.(e) {(x, y, z) ∈ R
3 : x2 + y2 ≤ a2 e y2 + z2 ≤ a2}, (a > 0).
(f) {(x, y, z) ∈ R3 : x ≤ z ≤ 1− y2 e x ≥ 0}.
(g) {(x, y, z) ∈ R3 : 4x+ 2y ≤ z ≤ 3x+ y + 1, x ≥ 0 e y ≥ 0}.
8. Utilizando integral dupla, calcule a area do conjunto B dado
(a) B = {(x, y) ∈ R2 : x3 ≤ y ≤ √
x}.(b) B e determinado pelas desigualdades xy ≤ 2, x ≤ y ≤ x+ 1 e x ≥ 0.
(c) B =
{
(x, y) ∈ R2 : x > 0,
4
x≤ 3y ≤ −3x2 + 7x
}
.
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