Lista Cálculo Integral

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Lista 2 - Matema´tica para Economia II
1 - Calcule a derivada das seguintes func¸o˜es:
(a) f(x) =
∫ x
0
√
1 + 2t dt (b) f(x) =
∫ x
1
ln t dt (c) f(x) =
∫ x
2
t2sen t dt
(d) f(x) =
∫ x
−1
1
t+ t2
dt (e) f(x) =
∫ 2
x
cos(t2) dt (f) f(x) =
∫ 10
x
tg t dt
(g) f(x) =
∫ x2
0
(t2 + 3) dt (h) f(x) =
∫ lnx
0
etdt (i) f(x) =
∫ x3+2x2
1
sen t dt
(j) f(x) =
∫ √x
−1
etsen t dt (k) f(x) =
∫ x
−x
t dt (l) f(x) =
∫ x3
−x2
et dt
2 - Seja f(x) = etgx +
∫ x3+x
0
√
t2 + 4 dt. Calcule f(0) e f ′(0).
3 - Seja f(x) = cos(g(x)) onde g(x) =
pi
2
+
∫ x2+3x
0
ln(e+ sen t) dt. Calcule f(0) e f ′(0).
4 - Dados
∫ 2
0
f(x) dx = −1,
∫ 6
2
f(x) dx = 10 e
∫ 6
2
g(x) dx = 3, calcule a integral defi-
nida.
(a)
∫ 6
2
[f(x)− g(x)] dx (b)
∫ 6
2
[2f(x)− 3g(x)] dx (c)
∫ 6
2
5f(x) dx (d)
∫ 6
0
f(x) dx
5 - Calcule as integrais abaixo.
(a)
∫ 1
0
(4x+ 1)2dx (b)
∫ 3
0
e2xdx (c)
∫ 2
1
1
x
dx (d)
∫ 4
1
−3√xdx
(e)
∫ 2
0
|2x− 1|dx (f)
∫ 1
−1
( 3
√
x− 2)dx (g)
∫ 4
1
x− 2√
x
dx (h)
∫ 4
0
1√
2x+ 1
dx
(i)
∫ 3
1
e3/x
x2
dx (j)
∫ 1
0
e2x
√
e2x + 1dx (k)
∫ 2
0
x
1 + 4x2
dx (l)
∫ 4
0
(2− |x− 2|)dx
(m)
∫ 2
1
x2exdx (n)
∫ 4
0
x
ex/2
dx (o)
∫ e
1
x5lnxdx (p)
∫ 0
−1
ln(1 + 2x)dx
6 - Esboce a regia˜o limitada pelos gra´ficos das func¸o˜es e determine sua a´rea.
1
(a) y = 1/x2, y=0, x = 1, x = 5 (b) y = 3
√
x, y = x
(c) y = x2 − 4x+ 3, y = 3 + 4x− x2 (d) y = 8/x, y = x2, y = 0, x = 1, x = 4
(e) y = ex/2, y = −1/x, x = 1, x = 2 (f) x = y2, x = y + 2
(g) x =
√
y, y = 9, x = 0 (h) y = 4√
x+1
, y = 0, x = 0, y = 8
7 - Calcule as integrais impro´prias:
(a)
∫ +∞
0
1
x
√
x
dx (b)
∫ +∞
0
1
(x+ 1)(x+ 2)
dx (c)
∫ +∞
1
lnx
x
dx
(d)
∫ +∞
−∞
|x|e−x2dx (e)
∫ 1
−∞
1
(2x− 3)2dx (f)
∫ +∞
−∞
x
x2 + 1
dx
(g)
∫ +∞
0
x senxdx (h)
∫ 4
0
1√
x
dx (i)
∫ 1
0
cos( 3
√
x)
3
√
x2
dx
(j)
∫ 1
1/2
1
x 7
√
(lnx)2
dx (k)
∫ 1
−1
1
x3
dx (l)
∫ 3
0
1
(x− 1)2dx
8 - Calcule as seguintes integrais duplas:
(a)
∫ ∫
R
sen(x+ y) dA, R = [0, pi/2]×[0, pi/2] (b)
∫ ∫
R
xy2
x2 + 1
dA, R = [0, 1]×[−3, 3]
(c)
∫ ∫
R
x sen(x+ y) dA, R = [0, pi/6]×[0, pi/3] (d)
∫ ∫
R
ye−xy dA, R = [0, 2]×[0, 3]
(e)
∫ ∫
R
y−2ex/
√
y dA, R = [0, 1]× [0, 2] (f)
∫ ∫
R
1
1 + x+ y
dA, R = [1, 3]× [1, 2]
9 - As integrais abaixo na˜o podem ser calculadas exatamente, em termos de func¸o˜es
elementares, com a ordem de integrac¸a˜o dada. Inverta a ordem de integrac¸a˜o e fac¸a os
ca´lculos.
(a)
∫ 1
0
∫ 1
y
ex
2
dxdy; (b)
∫ 1
0
∫ 1
x
sen y
y
dydx.
10 - Calcule as seguintes integrais duplas:
(a)
∫ ∫
R
xey dA, onde R e´ o triaˆngulo de ve´rtices (0, 0), (1, 0), (1, 1).
2
(b)
∫ ∫
R
(2y − x) dA, onde R e´ a regia˜o limitada por y = x3 e y = 2x.
(c)
∫ ∫
R
(2x+ 1) dA, onde R e´ o triaˆngulo de ve´rtices (−1, 0), (1, 0), (0, 1).
(d)
∫ ∫
R
1
y2 + 1
dA, onde R e´ o triaˆngulo limitado por y = x/2, y = −x e y = 2.
(e)
∫ ∫
R
12x2ey
2
dA, onde R e´ a regia˜o no primeiro quadrante limitada por y = x3 e
y = x.
(f)
∫ ∫
R
dA, onde R e´ a regia˜o limitada por y = lnx, y = 0 e x = e.
Gabarito:
1.
(a) f ′(x) =
√
1 + 2x (b) f ′(x) = ln x (c) f ′(x) = x2senx
(d) f ′(x) =
1
x+ x2
(e) f ′(x) = −cos(x2) (f) f ′(x) = −tgx
(g) f ′(x) = 2x(x4+3) (h) f ′(x) = 1 (i) f ′(x) = (3x2+4x)sen (x3+2x2)
(j) f ′(x) =
e
√
xsen
√
x
2
√
x
(k) f ′(x) = 0 (l) f ′(x) = 2xe−x
2
+ 3xex
3
2. f(0) = 1 e f ′(0) = 3 3. f(0) = 0 e f ′(0) = −3 4. (a) 7 (b) 11 (c) 50 (d) 9
5. (a) 31/10 (b) (e6− 1)/2 (c) ln 2 (d) −14 (e) 5/2 (f) −4 (g) 2/3 (h) 2 (i) (e3− e)/3
(j) 1
3
[(e2 + 1)3/2 − 2√2] (k) 1
8
ln 17 (l) 4 (m) e(2e− 1) (n) −12e−2 + 4 (o) 5
36
e6 + 1
36
(p) 2 ln 2− 1
6. (a) 4/5 (b) 1/2 (c) 64/3 (d) 7/3 + 8 ln 2 (e) (2e + ln 2) − 2e1/2 (f) 9/2 (g)
18 (h)16
7. (a) 2 (b) ln 2 (c) +∞ (d) 1/2 (e) 1/2 (f) diverge (g) na˜o existe (h) 4 (i)
3 sen(1) (j) 7/5(ln 2)5/7 (k) diverge (l) diverge
3
8. (a) 0 (b) 9 ln 2 (c)
1
2
(
√
3− 1)− pi
12
(d)
1
2
(e−6 +
5
2
) (e) 2(e− e1/
√
2 +
√
2
2
− 1)
(f) 6 ln 6− 5 ln 5− 4 ln 4 + 3 ln 3
9. (a) e/2− 1 (b) 1− cos 1
10. (a) 1/2 (b) 44/15 (c) 1 ,(d) 3/2 ln 5 (e) 2(e− 2) (f) 1
4