Math 122 - Makeup Exam - 3/10/2016
Section:
Name:
The following rules apply:
• This is a closed-book exam. You may not use
any books or notes on this exam.
Problem
Points
1
20
2
15
3
10
4
30
5
5
6
5
• You have 50 minutes to complete this exam.
When time is called, stop writing immediately and
turn in your exam to the nearest proctor.
7
5
8
5
• You may not use any electronic devices including (but not limited to) calculators, cell phone,
or iPods. Using such a device will be considered
a violation of the university’s academic integrity
policy and, at the very least, will result in a grade
of 0 for this exam.
9
5
Total:
100
• For free response questions, you must show
all work. Answers without proper justification
will not receive full credit. Partial credit will be
awarded for significant progress towards the correct answer. Cross off any work that you do not
want graded.
• For multiple choice questions, circle the letter of the best answer. Make sure your circles include just one letter. These problems will be
marked as correct or incorrect; partial credit will
not be awarded for problems in this section.
Score
Math 122
Exam 2 - Page 2 of 9
2/26/2016
Part I: Free Response
1. Consider the function f (x) = ln(x).
(a) (10 points) Set up but do not evaluate a Riemann Sum using right endpoints for the
area bounded between f and the x-axis over the interval [1, e2 ].
(b) (10 points) Set up and evaluate a definite integral to find the area bounded between f
and the x-axis over the interval [1, e2 ].
Math 122
Exam 2 - Page 3 of 9
2/26/2016
2. (15 points) At Charlie’s Chocolate Factory, a tank in the shape of an inverted right circular
cone has a height of 10 meters and a radius (at the top) of 6 meters is filled with chocolate
pudding to a height of 2 meters. In order to sterilize the tank, the factory needs to empty the
tank. Set up but do not evaluate an integral to find the work required to empty the tank by
pumping the chocolate pudding through a hole in the top of the tank. Note: the weight-density
of chocolate pudding is 12,178 N/m3 .
Math 122
Exam 2 - Page 4 of 9
Z
3. (10 points) Evaluate
2x2 + 7x − 3
dx.
x3 − 2x2 − 3x
2/26/2016
Math 122
Exam 2 - Page 5 of 9
2/26/2016
4. Suppose R is the region enclosed by y = ln x, y = 2, and x = 1.
(a) (4 points) On the plot given below, label all of the curves. Also, label all points at which
the curves intersect as ordered pairs (x, y).
(b) (6 points) Set up but do not evaluate an integral which represents the area of R by
integrating with respect to x.
(c) (6 points) Set up but do not evaluate an integral which represents the area of R by
integrating with respect to y.
Math 122
Exam 2 - Page 6 of 9
2/26/2016
(d) (8 points) Compute the area of R.
(e) (6 points) Suppose R is the base of a solid whose cross sections taken perpendicular to
the x-axis are squares. Set up but do not evaluate an integral which represents the
volume of this solid.
Math 122
Exam 2 - Page 7 of 9
2/26/2016
Part II: Multiple Choice
2
5. (5 points) Which of the following is the arc length of f (x) = (x − 1)3/2 from x = 0 to x = 1?
3
(a) 0
(b) 1
(c) 2
(d) 3/2
(e) 2/3
Z
6. (5 points) Evaluate the following improper integral:
1
(a) −∞
(b) −2
(c) 0
(d) 2
(e) +∞
∞
1
√ dx.
x
Math 122
Exam 2 - Page 8 of 9
2/26/2016
7. (5 points) Which of the following functions is a solution to the given initial value problem?

1
dy


=√


 dx
1 − x2


1


=π
 y
2
(a) y = sin−1 x +
5π
6
(b) y = sin−1 x + π
(c) y = − ln (1 − x) + π
(d) y = tan−1 x + π
(e) y = tan−1 x +
1
2
Z
8. (5 points) Suppose F (x) =
x
f (t) dt, where f (t) is shown below.
0
Which of the following quantities is the greatest?
(a) F (0)
(b) F (1)
(c) F (5)
(d) F (1) − F (5)
(e) F (5) − F (1)
Math 122
Exam 2 - Page 9 of 9
2/26/2016
9. (5 points) Which of the following integrals is/are improper?
Z
I.
1
(a) I only
(b) II only
(c) I and II only
(d) I and III only
(e) I, II, and III
∞
1
dx
x2
Z
1
II.
−1
1
dx
x
Z
III.
π/2
tan x dx
0