Math 126
Carter
Test 1
Spring 2013
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 114 total points available on the test. Slices of ripe avocado are a
pleasant addition to a sandwich.
1. (8 points each) Compute the following:
(a)
Z
(4x3 − 6x2 − 8x + 4) dx
(b)
Z π
sin (θ) dθ
0
(c)
Z 32
x−6/5 dx
1
(d)
Z
sin (4t) cos3 (4t) dt
(e)
Z 4
2
dx
x(ln (x))2
(f)
Z
sin (x)ecos (x) dx
(g)
Z
2y dy
y 2 + 16
(h)
Z 1
5−t dt
0
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=
j=1
(Hint: factor, don’t multiply.)
1
N (N + 1)
.
2
4. (10 points) Compute the volume of the cone that is obtained by rotating the triangle
that is bounded by y = 0, x = 0 and y = 10 − x/3 about the x-axis.
5. (10 points) Compute the volume that is obtained by rotating the area under the
parabola y = x2 between x = 2 and x = 5 about the y-axis.
6. (10 points) √
Compute the surface area of the sphere that is obtained by revolving the
curve y = 25 − x2 between x = −5 to x = 5 about the x-axis. Recall, that the
surface area formula is given by
A=
Z b
q
2πf (x) 1 + (f 0 (x))2 dx.
a
(Hint: the resulting integral is easy to calculate.)
7. (10 points) If a force of 90N ewtons stretches a spring 1 meter beyond its natural
length, how much work does it take to stretch the spring 5 meters beyond its natural
length? Recall that 1 Newton = 1 kilogram meter per square second, and that Hook’s
law states that the force required to stretch a spring is proportional to the length that
it is stretched.
2