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)
.
6
1
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.
2