# 234fall14FinalSol

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15 Feb 2019
School
Department
Course
Professor
Math 234 Final
Fall Quarter 2014
December 10, 2014
Your instructor: Xia (11:00am) Getzler (1:00pm)
Instructions:
Write legibly.
Show all your work on these sheets.
Make sure that your ﬁnal answer is clearly indicated. If two answers are presented then the average
of the points for each answer will be given!
This exam has 10 pages, and 9 problems. Before starting the exam, please check that your copy
contains all of them and obtain a new copy of the exam immediately if it does not.
You may not use books, notes or calculators.
Good luck!
Problem Points Score
possible
1 10
2 10
3 10
4 10
5 12
6 10
7 15
8 10
9 13
TOTAL 100
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Question 1 (10 points).
(1) Sketch the region Din the double integral
ZZD
f dA =Z2
0Z22x
x2
f(x, y)dy dx.
(2) Rewrite the double integral as an iterated integral with the order interchanged.
Solution:
Z2
0Z22x
x2
f(x, y)dy dx =Z4
0Zy
y2
8
f(x, y)dx dy.
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Question 2 (10 points).Evaluate the surface integral RRSpx2+y2dS, where Sis the spiral ramp
r(u, v) = ucos vi+usin vj+vk,
with 0 u1 and 0 vπ.
Solution:
ru(u, v) = cos vi+ sin vj,rv(u, v) = usin vi+ucos vj+k, , ru×rv= sin vicos vj+uk, .
ZZSpx2+y2dS =Zπ
0Z1
0
upu2+ 1dudv
=πZ1
0
upu2+ 1du
=π
3(u2+ 1)3
2
1
0=π
3(221).
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