Northwestern University

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Math 234-0 Final Exam

Winter Quarter 2012

Monday, March 12, 2012

Instructions:

•Read each problem carefully.

•Write legibly.

•Show all of your work on these sheets.

•The exam consists of 10 problems and 11 pages. Please verify that you have all 11

pages.

•You may not use books, notes or calculators.

•You have 120 minutes to complete this exam.

Problem Points Score

1 10

2 10

3 10

4 10

5 10

6 10

7 10

8 10

9 10

10 10

TOTAL 100

Double-angle formula:

cos2θ=1 + cos 2θ

2

sin2θ=1−cos 2θ

2

Math 234-0 Final Exam Winter Quarter 2012 Page 2 of 11

1. In evaluating a double integral over a region D, a sum of iterated integrals was

obtained as follows

Z2

0Z√2−y

2

−√2−y

2

f(x, y)dx dy +Z0

−1Z√1−y2

−√1−y2

f(x, y)dx dy.

(a) Sketch the region Dof integration.

(b) Express the double integral as one iterated integral with reversed order of

integration.

Math 234-0 Final Exam Winter Quarter 2012 Page 3 of 11

2. Evaluate the triple integral ZZZD

y2dV

where Dis the solid region bounded by the parabolic cylinder z= 1 −y2, and the

planes x+ 2z= 2, x= 0, and z= 0.