Math 234 Final
Fall Quarter 2014
December 10, 2014
Your instructor: Xia (11:00am) Getzler (1:00pm)
•Read each problem carefully.
•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.
Problem Points Score
Question 1 (10 points).
(1) Sketch the region Din the double integral
f dA =Z2
f(x, y)dy dx.
(2) Rewrite the double integral as an iterated integral with the order interchanged.
f(x, y)dy dx =Z4
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.
ru(u, v) = cos vi+ sin vj,rv(u, v) = −usin vi+ucos vj+k, , ru×rv= sin vi−cos vj+uk, .