MATH 152, SPRING 2012
COMMON EXAM II - VERSION A
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Section No:
PART I: Multiple Choice (4 pts each)
1.
Z
1
dx =
x2 (x − 1)
1
+C
x
ln |x2 (x − 1)| + C
1
ln |x| − − ln |x − 1| + C
x
1
− ln |x| + + ln |x − 1| + C
x
1
ln |x − 1| − + C
x
(a) ln |x − 1| +
(b)
(c)
(d)
(e)
2. The improper integral
Z
1
e
dx
x ln x
(a) diverges to ∞.
(b) diverges to −∞.
(c) converges to 1.
(d) converges to −1.
1
(e) converges to − 1.
e
3. Which of the following integrals gives the surface area obtained by rotating the curve y = e−4x , for 0 ≤ x ≤ 1,
about the y-axis?
Z 1
p
(a)
2πe−4x 1 + 16e−8x dx
0
1
(b)
Z
e−4
(c)
Z
1
(d)
Z
1
(e)
Z
p
2πx 1 + 16e−8x dx
0
1
0
0
r
2πy 1 +
1
dy
16y 2
πp
16y 2 + 1 dy
2
π ln y p
16y 2 + 1 dy
8 y
1
4. Compute
Z
∞
−1
dx
.
1 + x2
3π
4
π
(b)
2
π
(c)
4
(d) ∞
(a)
(e) 0
5. By substituting x = 3 tan θ, the integral
Z
0
π/4
(a)
Z
3
(b)
Z
Z
π/4
(c)
π/4
(d)
Z
Z
π/4
3
x2
p
x2 + 9 dx becomes
27 tan2 θ sec θ dθ
0
27 tan2 θ sec3 θ dθ
0
81 tan3 θ sec2 θ dθ
0
81 tan2 θ sec2 θ dθ
0
(e)
81 tan2 θ sec3 θ dθ
0
6.
∞ (−1)n + 2n
P
=
6n
n=0
(a)
(b)
(c)
(d)
(e)
1
3
5
14
33
14
3
10
27
10
2
7. Find the length of the curve x = t2 , y = t3 , for 0 ≤ t ≤ 1.
(a)
(b)
(c)
(d)
(e)
1 √
13 13 − 8
27
2π √
13 13 − 8
27
1 √
13 13 − 1
27
1
27
2π √
13 13 − 1
27
8. Which of the following series diverges by the Test for Divergence?
∞ ln n
P
n=1 n
∞
P
π
1
(b)
sin
−
2
n
n=1
∞
P n
(c)
n=1 n!
∞
P
1
(d)
sin
n
n=1
(a)
(e) The Test for Divergence is inconclusive for all of the above series.
4
converges. Find the limit.
an
9. The recursive sequence defined by a1 = 2, an+1 = 5 −
(a) 1
(b) 4
(c) 5
5
(d)
2
(e) 2
10. Which of the following statements is true regarding the improper integral
Z
1
Z
∞
Z
∞
Z
∞
∞
dx
√ ?
ex + x
dx
dx
dx
√ <
√ and
√ converges.
(a) The integral converges because
x
e + x
x
x
1
1
1
Z ∞
Z ∞
Z ∞
dx
dx
dx
√ and
√ diverges.
√ >
(b) The integral diverges because
ex + x
x
x
1
1
1
Z ∞
Z ∞
Z ∞
dx
dx
dx
√ >
(c) The integral diverges because
and
diverges.
ex
ex
ex + x
1
1
1
Z ∞
Z ∞
Z ∞
dx
dx
dx
√
(d) The integral converges because
<
and
converges.
ex
ex
ex + x
1
1
1
(e) The integral converges to 0.
3
11. The sequence whose terms are an =
n2 − 1
n2
(a) decreases and converges to 1.
(b) increases and converges to 1.
(c) decreases and converges to 0.
(d) increases and converges to 0.
(e) diverges.
12. Find the surface area obtained by rotating the curve x = cos(2t), y = sin(2t), for 0 ≤ t ≤
π
4
(b) 2π
π
(c)
2
(d) π
(a)
(e) 4π
13. Find the sum of the geometric series S =
4
8
16
+
+
+ ....
9 27 81
(a) S = 3
(b) S = 2
2
(c) S =
3
4
(d) S =
15
4
(e) S =
3
4
π
, about the x-axis.
4
PART II: WORK OUT (52 points total)
Directions: Present your solutions in the space provided. Show all your work neatly and concisely and box your
final answer. You will be graded not merely on the final answer, but also on the quality and correctness of the work
leading up to it.
Z p
14. (10 pts) Integrate
16 − 9x2 dx.
π
π
cos − cos
15. (8 pts) Find the sum of the series: S =
n
n+1
n=1
∞
P
5
16. (10 pts) Integrate
Z
4x2 − 1
dx.
(x2 + 1)(x − 2)
6
17. If the nth partial sum of the series
∞
P
an is given by sn =
n=1
(i) (5 pts) Find a10 .
(ii) (5 pts) Find the sum of the series S =
∞
P
2n + 1
,
n
an .
n=1
18. (10 pts) Find the surface area obtained by rotating the curve y =
7
1
x2
− ln x, for 1 ≤ x ≤ 2, about the y-axis.
4
2
Last Name:
First Name:
Section No:
MATH 152, SPRING 2012
COMMON EXAM II - VERSION A
Question
Points Awarded
Points
1-13
52
14
10
15
8
16
10
17
10
18
10
100
8