# Math 234 Winter 2012 Final

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15 Feb 2019
School
Department
Course
Professor
Northwestern University
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Math 234-0 Final Exam
Winter Quarter 2012
Monday, March 12, 2012
Instructions:
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=1cos 2θ
2
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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
0Z2y
2
2y
2
f(x, y)dx dy +Z0
1Z1y2
1y2
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.
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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.
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