Math 234 Final

Fall Quarter 2014

December 10, 2014

Your Name

Your instructor: Xia (11:00am) Getzler (1:00pm)

Instructions:

•Read each problem carefully.

•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

Question 1 (10 points).

(1) Sketch the region Din the double integral

ZZD

f dA =Z2

0Z2√2x

x2

f(x, y)dy dx.

(2) Rewrite the double integral as an iterated integral with the order interchanged.

Solution:

Z2

0Z2√2x

x2

f(x, y)dy dx =Z4

0Z√y

y2

8

f(x, y)dx dy.

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 ≤u≤1 and 0 ≤v≤π.

Solution:

ru(u, v) = cos vi+ sin vj,rv(u, v) = −usin vi+ucos vj+k, , ru×rv= sin vi−cos vj+uk, .

ZZSpx2+y2dS =Zπ

0Z1

0

upu2+ 1dudv

=πZ1

0

upu2+ 1du

=π

3(u2+ 1)3

2

1

0=π

3(2√2−1).