# Name: Math 2260 Final Exam May 2, 2012

```Name:
Math 2260 Final Exam
May 2, 2012
Instructions: You are welcome to use one sheet of notes, but no other references or tools are allowed
(no textbooks, no calculators, etc.). This is a 2 hour exam; you may start working at 8:00 am and must
stop at 10:00 am. To receive full credit for a correct answer you must demonstrate how you arrived at
1. (5 points) Consider the region bounded above by the graph of y = sin(x/2) and below by the x-axis.
What is the volume of the solid obtained by revolving this region around the x-axis?
2
2. (5 points) Find the surface area of the cone given by rotating the line y = 2x for 0 ≤ x ≤ 4 around the
x-axis.
3
3. (5 points) Solve the initial-value problem
dy
− xy = x,
dx
4
y(0) = 3.
4. (5 points) Evaluate the integral
Z
e3
1
5
ln x
√
dx.
3
x2
5. (5 points) Evaluate the integral
Z
2
√
0
6
1
dx.
4 − x2
6. (8 points)
(a) Does the series
∞
X
(−1)n
n=1
n2 + π
√
n3 − 2/2
converge absolutely, converge conditionally, or diverge?
(b) Does the series
∞
X
(n + 3)3
(n + 1)!
n=1
converge absolutely, converge conditionally, or diverge?
7
7. (5 points) What is the interval of convergence of the following power series?
∞
X
(−1)n (x − 2)n
.
n5n
n=1
8
8. (5 points) Find an approximation for
Z
0
1
sin(t2 )
dt
t
which is accurate to within 0.01. Feel free to give your answer as a fraction.
9
*
*
*
*
*
9. (5 points) u = h1, 2i and v = h4, 2i are two vectors in the plane. Find vectors a and b so that a is
* *
*
parallel to v, b is perpendicular to v, and
*
*
*
u = a + b.
*
u
*
*
v
b
*
a
10
```