Name:
APPM 2350 Exam #3 Version B Summer 2008
Be sure to include your name and a grading table on the front of your blue book. You must
work all of the problems on this exam. Show ALL of your work and BOX IN YOUR
FINAL ANSWERS. A correct answer with no relevant work may receive no credit, a
wrong answer with no work will receive no credit, and an incorrect answer accompanied
by some correct work may receive partial credit. Text books, class notes, crib sheets, cell
phones, calculators, or electronic devices of any kind are NOT permitted. Please start of
each new problem on a new page. Good luck!
1. (24 points) Integration!
(a) Evaluate
Zπ/2
π/6Z1
0
(1 −r2) cos θ
sin θr dr dθ
(b) The following double integrals represent the area of a region in the xy-plane.
Z0
−1Z√1+x
0
dy dx +Z1
0Z√1+x
√2x
dy dx
i. Sketch the region of integration.
ii. Rewrite the area of the region as one double integral.
iii. Evaluate the one double integral.
(c) Evaluate
ZZZRpx2+y2+z2dV
where Ris region bounded below by the cone φ=π
3and above by the sphere
ρ= 2.
(d) Evaluate
Z2
0Z√2−x2
0Z√x
0
y z dy dz dx
2. When studying the spread of an epidemic, we can model the likelihood a disease will
spread to an individual. Suppose that our model indicates a person has over a 50%
chance of contracting a disease if they are within 1 mile of 100 other infected individuals.
Assume the population density of a circular city centered at the origin is approximated
by
δ(x, y) = 50
(1 + x2+y2)2individuals per square mile
Will a person located at the center of the city have a 50% chance or greater of con-
tracting the disease?
