Download Improper Integrals, Arclength, Surface Area, Centroids

Improper Integrals, Arclength, Surface Area, Centroids
I. Decide if the following are convergent or divergent integrals. Evaluate them if they are convergent.
Z
5
dx
3x − 6
2
Z ∞
dx
2
x +4
0
Z 1
ln(x) dx
√
6 (convergent)
(1)
=
π
(convergent)
4
(2)
=
−1 (convergent)
(3)
=
π
(convergent, u = ex )
4
(4)
=
0 (convergent)
(5)
(divergent)
(6)
0
∞
ex
dx
e2x + 1
Z0 ∞
x
dx
4+1
x
−∞
Z ∞
xex dx
Z
=
0
II. Use the Comparsion Test to decide if the following converge or diverge:
∞
ex
√
x
1
Z ∞ √
x
2 + 1)
(x
2
Z ∞
cos2 (x)
x2 + 2
Z ∞0
arctan(x)
x3 + 2
0
Z ∞
sin2 (x) + 1
√
x+1
1
Z
dx
dx
dx
dx
dx
ex
1
√ ≥ √ f orx > 1
x
x
√
√
x
x
1
converges 0 ≤ 2
≤ 2 = 3/2 f orx > 2
(x + 1)
x
x
cos2 (x)
1
1
converges 0 ≤ 2
≤ 2
≤ 2 f orx > 0
x +2
x +2
x
arctan(x)
π 1
π 1
converges 0 ≤
≤
≤
f or x > 0
x3 + 2
2 x3 + 2
2 x3
sin2 (x) + 1
1
1
√
diverges
≥√
≥ √ f or x ≥ 1
x+1
x+1
2 x
diverges
(7)
(8)
(9)
(10)
(11)
III. Find the arclength of the following curves:
1
y = (1 + 2x)(3/2)
3
y=
1 2
(x + 2)(3/2)
3
Z
y=
0
x
√
et − 1 dt
0<x<1
1
AN S : L =
0
Z
√
√
8 2 2
2 + 2x dx = −
3
3
AN S : L =
1<x<8
AN S :
1<x<2
AN S :
0 < x < π/6
AN S :
0 < x < ln(4)
AN S :
0
(12)
1
4
(1 + x2 ) dx =
3
0
Z 8 √ 2/3
x +2
L=
dx = 33/2 (23/2 − 1)
1/3
x
1
Z 2
4 + x6
21
L=
dx =
3
4x
16
1
Z π/6
1
L=
sec(x) dx = ln(3)
2
0
Z ln(4)
L=
2ex/2 dx = 2
0<x<1
3
y = √ x(2/3)
2
x6 + 8
y=
16x2
y = log(sec(x))
Z
(13)
u = x2/3(14)
(15)
(16)
(17)
IV. Find the surface area of the surface formed by revolving the following curves about the x-axis:
Z 3
√
√
√
4π 3 x dx = 16π 3
y = 3x 1 < x < 3 AN S :
S=
(18)
1
y=
√
1
<x<1
3
AN S :
2 3
x
3
0<x<1
AN S :
25 − x2
2<x<3
AN S :
3x + 1
y=
y=
p
−
1
√
49
π 12x + 13 dx =
π
9
−1/3
Z 1
4 3p
1 3/2
5 −1 π
πx 1 + 4x2 dx =
S=
3
18
0
Z 3
S=
10π dx = 10π
Z
S=
(19)
(20)
(21)
2
y = cos(x)
y=
3 2/3
x
2
π/4
Z
π
0<x<
4
AN S :
0<x<1
AN S :
p
2πcos(x) 1 + sin2 (x) dx
S=
1
Z
3πx1/3
S=
0
p
6 √
1 + x2/3 dx =
2 + 1) π
5
u = x1/3
Z 1
p
S=
πx2 1 + x2 dx
use
1
y = x2
2
0<x<1
(22)
0
AN S :
0
(23)
(24)
√ π √
=
ln( 2 − 1) + 3 2 (25)
8
8. [10pts] Find the x and y coordinates of the center of mass of the centroid (region) bounded by the curves:
y=
1 2
x , y = 0, x = 0, x = 2,
2
y = ex , y = 0, x = 0, x = ln(2)
y = sin(x), y = 0, x = 0, x =
π
2
x̄ =
3
3
, ȳ =
2
5
x̄ = 2 ln(2) − 1, ȳ =
0<x<
x̄ = 1 , ȳ =
3
4
π
8
Indefinite integrals and Trigonometric Identities
Some commonly used trigonometric identities are:
sin2 (x) + cos2 (x)
=
1
tan (x) + 1
=
cos2 (x)
=
sin2 (x)
=
sin(2x)
=
sin(x) cos(y)
=
cos(x) cos(y)
=
sin(x) sin(y)
=
sec2 (x)
1
(1 + cos(2x))
2
1
(1 − cos(2x))
2
2 sin(x) cos(x)
1
(sin(x + y) + sin(x − y))
2
1
(cos(x + y) + cos(x − y))
2
1
(cos(x − y) − cos(x + y))
2
2
Some commonly integrals worth noting include:
Z
1
u2 + 1
Z
1
√
1 − u2
Z
tan(u)
Z
sec(u)
du
= arctan(u) + c
du
= arcsin(u) + c
du
= ln|sec(u)| + c
du
= ln |sec(u) + tan(u)| + c