Math 1206
Common Final Exam
Fall 2011
Form C
Instructions: Please enter your NAME, ID Number, FORM DESIGNATION, and your CRN on the opscan sheet. The CRN should be written in the box labeled ‘COURSE‘. Do not include the course number.
Darken the appropriate circles below your ID number and below the Form designation letter. Use a
number 2 pencil. Machine grading may ignore faintly marked circles.
Mark your answers to the test questions in rows 1 - 14 of the op scan sheet. Your score on this test will be
the number of correct answers. You have one hour to complete this portion of the exam. Turn in the op
scan sheet with your answers and the question sheets, including this cover sheet, at the end of this part
of the final exam.
Exam Policies: You may not use a book, notes, formula sheet, or a calculator or computer. Giving or
receiving unauthorized aid is an Honor Code Violation.
Signature:
Name (printed)
Student ID #
1. A tank in the form of an inverted right circular cone is full of water. If the height of the tank is 10 ft
and the radius of its top lid is 4 ft, determine an integral that represents the work done in pumping
all the water over the lip of the tank. The density of the water is δ = 62.4 lb/ft3 .
H0,0,10L
water
r=4
A. 2πδ
H0,0,0L
4
25
R4
0
x2 (x − 4) dx
4 R4 2
B. 2πδ
0 x (4 − x) dx
25
4 R 10 2
C. πδ
0 y (y − 10) dy
25
4 R 10 2
D. πδ
0 y (10 − y) dy
25
1
2. A solid is generated by revolving about the y-axis the region in the first quadrant bounded above
by the circle x2 + y 2 = 9 and below by the ellipse 4x2 + 9y 2 = 36. The integral for the volume of the
solid is:
3
Z
A. 2π
p
x
9 − x2 − 2
r
1−
0
Z
3
9−y
C. π
2
0
x 2
!
Z
dx
3
3
p
x
9 − x2 − 2
B. 2π
r
1−
−3
9 2
dy
− 9− y
4
Z
3
D. π
2
x 2
3
9 2
(9 − y ) − 9 − y
dy
4
2
2
3. Find the length of the segment of the curve y = (x2 + 1)3/2 from x = 1 to x = 4.
3
A. 48
4. Find
d
dc
B. 45
Z
C.
4527
5
D. 3 −
y
cos(yx) dx.
4c
A. 4 cos(4cy)
Z
1
5. Evaluate
0
A.
x2
B. 4 sin(4cy)
C.
4
sin(4cy)
y
1
ln(5) + ln(2) − ln(6)
4
6. Find the area of the region bounded by y =
19
3
D. −4 cos(4cy)
1
dx.
+ 6x + 5
C. ln(12) − ln(5)
A.
4√
34 √
2+
17
3
3
B.
26
3
C.
B.
1
ln(2) − ln(6)
4
D.
1
1
−
12 5
1
, y = x, and x = 3.
x2
10
3
D. 4
2
!
dx
Z
7. If there is a substitution y(x) so that
1
A. A = √
2
B. A =
√
1
dx = A tan−1 y(x) + c, find A.
x2 + 2x + 3
2
1
2
C. A =
Z
1
8. Using Simpson’s rule with 4 subintervals to evaluate
1
D. A = √
3
2
e−x dx, the value would be
−1
1 −1
e + 2e−1/4 + 2 + 2e−1/4 + e−1
4
1 −1
e − 4e−1/4 + 2 − 4e−1/4 + e−1
C.
6
1 −1
e + 4e−1/9 + 2e−1/9 + 4e−1
6
1 −1
D.
e + 4e−1/4 + 2 + 4e−1/4 + e−1
6
A.
B.
9. Determine whether each of the following limits exists:
x + sin(x)
x→0
ex − 1
(1)
A.
(1) does not exist.
(2) exists.
B.
lim
(2)
(1) exists.
x + sin(x)
x→∞
ex − 1
lim
C. Both limits exist.
(2) does not exist.
D. Neither limit exists.
10. Find the lower Riemann sum of the distance traveled by a model train engine moving along the
track for 5 seconds.
Time (sec.)
Velocity (in/sec.)
0
0
1
12
2
22
3
10
4
5
5
8
A. 5 [12 + 22 + 22 + 10 + 8]
B.
[12 + 22 + 20 + 5 + 8]
C. 5 [0 + 12 + 22 + 10 + 5]
D.
[0 + 12 + 10 + 5 + 5]
3
11. When the proper trigonometric substitution is made for evaluating the integral
Z
√
x2
dx,
1 − 4x2
the resulting integral is
1
8
Z
1
C.
8
Z
A.
Z
sin2 (θ) dθ + C
1
8
Z
sin2 (θ)
dθ + C
cos(θ)
1
D.
8
Z
sec3 (θ) dθ + C
B.
2
tan (θ) sec(θ) dθ + C
π/2
12. Evaluate
sin2 (3x) cos(3x) dx.
0
A. −
1
2
B. −
1
9
C. −
13. Evaluate
14. Evaluate
A. 3
Z
1
3
D.
π3
72
3x3
dx
(x4 + 5) ln(x4 + 5)
A.
3
[ln(x4 + 5)]2 + C
8
B.
3
ln[ln(x4 + 5)] + C
4
C.
3
ln(x4 + 5) + C
4
D.
ln[ln(x4 + 5)] + C
R∞
0
e−3x dx.
B. −
1
3
C.
1
3
D. The integral diverges.
4