Math 126
Carter
Study Guide Test 1
Spring 2013
First, here are a few tips.
1. I can’t test you on everything, but you should be prepared for anything.
2. On this test, I won’t ask about volumes of solids rotated about anything other than
the x, or y-axis. On the questions that I ask, they may be easier using disks, washers,
or cylindrical shells. Be familiar with the all of problems that I indicate below. Try to
construct similar problems.
3. I have indicated a large number of integrals that involve substitutions. Know all of
these problems! Also be able to work a simple problem such as the first one indicated
on the first test from Spring 2011 which appears below.
4. The format of the exam and length will be comparable to the test from Spring 2011.
However, we do not yet know how to solve #1e. I won’t ask a problem from section
7.2 (problem 4 on that test).
Here are some specific questions.
1. The integrals on the test will be chose from problems that are very similar to
• Section 5.4, p. 325 # 1-30;
• Section 5.5, p. 333 # 17,19,21,23,24,25,31,33,35,51,55;
• Section 5.6, p. 341 # 21,23,27,29,31,39;
• Section 7.1, p. 410 # 3,5,7,9,11,13,17,19,27;
The integrals will be intermingled among the sections, but I will make an effort to
make the problems move from easy to difficult.
2. Pay attention to # 2 and # 3 below. The variations on # 3 are to prove
•
n
X
!
j=
n(n + 1)
2
!2
j3 =
n(n + 1)
2
j=1
•
n
X
j=1
•
n
X
(2j − 1) = n2
j=1
Proofs for can be found at
http://www.youtube.com/watch?v=-usK5CMpUPo&list=ECB50BA7F6469AE91C
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3. For volumes pay attention to
• Section 6.1, p. 362 # 15,17,18,21;
• Section 6.2 p. 369 # 1,8,29;
4. For Arc-length and surface area pay attention to
• Section 6.3, p. 376 # 3,5,9;
• Section 6.4, p. 381 # 9;
• The arc length of a sector of a circle;
• The surface area of a slice of a sphere;
5. For other applications pay attention to
• (Work) Section 6.5, p. 386 # 1,4,5,7,8;
• (centers of mass) Section 6.6, p. 397 # 1, 15, 17.
The material below is the most recent exam that I administered in Calculus 2.
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Math 126
Carter
Test 1 Spring 2011
General Instructions: Write your name on only the outside of your blue book. Do all
of your work and write your answers inside your blue book. Put your test paper inside your
blue book as you leave. Solve each of the following problems; point values are indicated on
the problems. There are 106 total points available on the test. A tall glass of iced tap water
is consistently refreshing.
1. (8 points each) Compute the following:
(a)
Z
(3x2 + 2x + 1) dx
(b)
d Z x t2
e dt
dx 0
(c)
Z 1
(4x + 3)3 dx
0
(d)
Z 2
√
0
x2
dx
x3 + 1
(e)
Z
ex cos (ex ) dx
(f)
Z
dx
9 + x2
(g)
(ln (x))4
dx
x
Z
2. (5 points) Complete the statement of the fundamental theorem of calculus that begins:
“Assume that f (x) is a continuous function on the open interval (a,b), and let F (x)
be an anti-derivative of f (x) on the closed interval [a, b], then [. . .] .”
3. (5 points) Use the technique of proof by induction, to show that
N
X
j=1
j2 =
N (N + 1)(2N + 1)
.
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4. (10 points) The population of a city as a function of time is given by the formula
P (t) = 2 · 106 e0.06t .
When will the population double? You may leave your answer in terms of logarithms.
5. (10 points) Consider the line y = 2x + 3 and the parabola y = x2 . Compute the area
that is bounded between these two functions.
6. (10 points) Compute the volume that is obtained by rotating around the x-axis the
region in the plane that is bounded between y = x2 + 2 and y = 10 − x2 .
7. (10 points) Compute the volume of the solid that is obtained by rotating √
the region
2
that is above the x-axis and underneath the graph y = sin (x ) for x ∈ [0, π] about
the y-axis.
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