# Math 126 Carter Test 1 Spring 2011

```Math 126
Carter
Test 1 Spring 2011
General Instructions: Write your name on only the outside of your blue book. Do all
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 &middot; 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
```