MATH 172.504
Examination 1
February 14, 2012
NAME
SIGNATURE
This exam consists of 9 problems, numbered 1–9. For partial credit you must present your
work clearly and understandably and justify your answers.
The use of calculators is not permitted on this exam.
The point value for each question is shown next to each question.
CHECK THIS EXAMINATION BOOKLET BEFORE
YOU START. THERE SHOULD BE 9 PROBLEMS ON
7 PAGES (INCLUDING THIS ONE).
Do not mark in the box below.
1–5
6
7
8
9
Total
Points
Possible Credit
30
10
24
24
8
96
NAME
MATH 172
Examination 1
Page 2
Multiple Choice: [6 points each] In each of Problems 1–5, circle the best answer.
Z
1.
Evaluate
2
√
6x 2x2 + 1 dx.
0
(A) 26
(B) 27
(C) 28
(D) 29
(E) 30
2.
Consider the region in the first quadrant bounded by y = x2 , y = 9, and x = 0.
Which integral below gives the volume of the solid obtained by rotating this region
about the x-axis?
9
Z
(9 − x2 ) dx
(A) π
0
3
Z
x(9 − x2 ) dx
(B) 2π
0
Z
(C) 2π
9
(x4 − 9) dx
0
Z
3
(81 − x4 ) dx
(D) π
0
Z
(E) π
3
y 2 dy
0
February 14, 2012
NAME
3.
MATH 172
Examination 1
Page 3
Our ideal spring follows Hooke’s Law. It requires 5 Newtons of force to be held 10 cm
from rest. How much work (in Joules) is done in stretching the spring from rest to
20 cm?
(A) 0
(B) 1
(C) 10
(D) 100
(E) 1000
Z
4.
Let F (x) =
x
√
t3 + 1 dt. Which of the following statements about F (x) are true?
1
I. F (1) = 0.
II. F 0 (2) = 2.
III. F 0 (2) = 3.
(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I and III only
February 14, 2012
NAME
5.
MATH 172
Examination 1
Page 4
What is the average value of f (x) = cos(3x) on the interval 0 ≤ x ≤ π2 ?
(A) −
2
3π
(B) −
3
2π
(C)
1
3π
(D)
1
2
(E) 0
6.
[10 points] Let R be the region in
√ the first quadrant of the xy-plane bounded above
by the union of the curve y = 2 x and the line y = −x + 8 and bounded below by
the x-axis. Find the area of R by setting up the integral with respect to y.
February 14, 2012
NAME
7.
MATH 172
Examination 1
Page 5
[24 points] Evaluate the following integrals.
Z
(a)
π/3
4 cos2 θ sin θ dθ
0
Z
√
x2 x + 1 dx
Z
e2x
dx
e2x + 1
(b)
(c)
February 14, 2012
NAME
8.
MATH 172
Examination 1
Page 6
[24 points] Let R be the region in the first quadrant of the xy-plane bounded by the
curves
πx , y = 2 − 2x2 , and the y-axis.
y = cos
2
Set up integrals with respect to x (do not evaluate!) that express the volumes of the
solids obtained by rotating R about the following axes of rotation. (It may help to
start by sketching the region R.)
(a) The x-axis.
(b) The y-axis.
(c) The line y = −1.
February 14, 2012
NAME
9.
MATH 172
Examination 1
Page 7
[8 points] Let S be the 3-dimensional solid whose base is the region in the first
quadrant of the xy-plane bounded by y = cos x, the x-axis, and the y-axis, and whose
cross-sections perpendicular to the x-axis are squares. Write down an integral for the
volume of S. (You do not need to evaluate the integral.)
There is more space on this page than you need for this problem. Feel free to use
extra space here and on the back of this page for scratch work.
February 14, 2012