Math 126-104
Final Exam
Spring 2013
Carter
General Instructions: Write your name on only the outside of your blue book. Do all your work
inside your blue book. Write neat, complete, solutions to the problems below. Please label the
problem upon which you are working, and write a statement or paraphrase of the problem. Indicate
your solution clearly. There are 160 points. Breakfast cereal with yogurt, instead of milk, can be
a welcome change of pace.
1. Compute the following integrals.
(a)
1
Z
(x2 +
√
x) dx
0
(b)
π/2
Z
sin3 x dx
0
(c)
Z
ln (x) dx
(d)
Z
√
1
dx
+ 25
x2
(e)
Z
1
dx
(x + 2)(x − 1)
(f)
Z
∞
(x−6/5 ) dx
1
2. Compute the volume of the solid that is obtained by rotating about the y-axis the region
between the lines x = 1 and x = 2, above the x-axis, and below the curve y = x2 .
3. A leaky bag of sand that initially weighs 144 pounds and looses half of its weight when being
lifted from a height of 0 feet to a height of 18 feet. Compute the work done in lifting the bag.
4. Sum the series:
0 1 2
3
3
3
−
+ −
+ −
+ ···
5
5
5
5. Determine if the given series converges.
(a)
∞
X
1
n!
n=1
(b)
∞ X
5 n
1+
n
n=1
(c)
∞
X
1
n n2 − 1
n=2
√
6. Compute the interval of convergence for the series
∞
X
(x − 2)n
n=0
7. Use the geometric series
∞
X
10n
yn =
n=0
1
1−y
which converges for |y| < 1, use substitution (y = −x), and term-by-term integration to
obtain a series for
f (x) = ln (1 + x)
which is valid for |x| < 1.
8. Sketch the graph of the function given in polar coordinates that is given by
r = 1 + cos (θ)
9. Give a parametric representation for the ellipse
(x − 2)2 (y − 5)2
+
=1
9
36
that starts at the point (5, 5) and travels once around in a period t ∈ [0, 2π].