MATH 172.503
Examination 1
February 18, 2010
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
6 PAGES (INCLUDING THIS ONE).
Do not mark in the box below.
1–5
6
7
8
9
Total
Points
Possible
35
12
25
12
16
100
Credit
NAME
MATH 172.503
Examination 1
Page 2
Multiple Choice: [7 points each] In each of Problems 1–5, circle the best answer.
1.
Evaluate
Z
π
4
0
(A)
cos(2x)
dx.
1 + sin(2x)
1
ln(2)
2
(B) ln(2)
(C)
1
2
(D) 1
(E) 0
2.
Evaluate
Z
0
√
3
√
5x
dx.
x2 + 1
(A) 1
(B)
1
3
(C)
35
3
(D) 5
(E) 10
February 18, 2010
NAME
3.
MATH 172.503
Examination 1
Page 3
If a spring is pulled far enough, it can become distorted and no longer exhibit the
properties of a perfect spring. Suppose that pulling a particular spring a distance
x ≥ 1 meters requires a force (in Newtons) of
F (x) =
2 ln x
x
to have the spring remain stretched. How much work (in Joules) is done in pulling
the spring from 2 m to 3 m from rest?
(A) ln(2)
(B) ln(3)
(C) ln(3) − ln(2)
(D) ln(9) − ln(4)
(E) (ln(3))2 − (ln(2))2
4.
What is the average value of the function f (x) = cos3 (x) on the interval 0 ≤ x ≤ π2 ?
(A)
1
π
(B)
2
π
(C)
4
3π
(D)
1
2
(E)
2
3
February 18, 2010
NAME
5.
MATH 172.503
Let F (x) =
Z
3
2x
√
Examination 1
Page 4
1
dt. What is F ′ ( 12 )?
2
t + 3t
(A) 0
(B) 1
(C) 2
6.
(D)
1
2
(E)
2
3
[12 points] Find the area of the region in the plane which is bounded by the curves
x = 1 − y 2 and x = y 4 − y 2 − 15.
February 18, 2010
NAME
7.
MATH 172.503
Examination 1
Page 5
[25 points] Evaluate the following indefinite integrals.
(a)
Z
(b)
Z
x cos(3x) dx
(c)
Z
sin2 (x) dx
x5
dx
x3 − 1
February 18, 2010
NAME
MATH 172.503
Examination 1
Page 6
8.
[12 points] Find the volume obtained by rotating the region which is bounded above
by y = 4 − x2 and below by the x-axis about the line y = −1. (You do not need to
simplify your numerical answer completely.)
9.
[16 points] Let A be the first point to the right of the origin where the curves y =
sin(2x) and y = sin(x) intersect. Consider the region bounded by these two curves
between the origin and A.
Set up integrals with respect to x (do not evaluate!) that express the volumes obtained
by rotating this region about the following axes of rotation.
(a) The x-axis.
(b) The y-axis.
February 18, 2010