Spring 2016
MATH 150b
Exam 2
Directions:
•
DO NOT OPEN THIS EXAM UNTIL TOLD TO DO SO.
•
Show an appropriate amount of work for each problem, so that the reader can see not only your answer
but the reasoning you used and steps you took to obtain it.
•
You are free to use results from the assigned reading or lecture notes. You may not use results that haven’t
been discussed in class or in the assigned reading, unless you include a proof of these results as well.
•
You may not use electronic devices of any sort.
Points Earned
Points Possible
Problem 1
17
Problem 2
10
Problem 3
10
Problem 4
18
Problem 5
18
Problem 6
7
Total
80
NAME:
SOLUTIONS

1. Let
R
be the region in the first quadrant that is bounded by the curves
x
=
√
y
,
x
=
e
y
,
y
= 0, and
y
= 1.
(See the accompanying figure.)
(a) Determine the area of
R
.
0.5
1.0
1.5
2.0
2.5
3.0
x
0.2
0.4
0.6
0.8
1.0
y

(b) Let
S
be the solid obtained by rotating
R
around the
x
-axis. By using any method, set up an integral
that equals the volume of
S
. Do not evaluate the integral.

2. Let
Q
be the region in the plane given by the inequality
x
2
+
y
2
≤
4; in other words,
Q
is the region inside
of the circle
x
2
+
y
2
= 4.
Let
T
be the solid obtained by rotating
Q
around the line
y
= 5. By
using the method of washers
, set up
an integral that equals the volume of
T
. Do not evaluate your integral.