APPM 2350 Spring 2006 Exam 3 Apr 19, 2006 1
INSTRUCTIONS: Computers, calculators, books, notes, flailing monkeys, etc. are not
permitted. Some (possibly) useful formulae are attached. Write your name, your instruc-
tor’s name and your recitation section (and/or TA’s name) on the front of your bluebook.
Work all problems. Start each problem on a new page. Unless otherwise stated, show
all your work clearly and box your final answer; a correct answer with incorrect or no
supporting work will receive no credit, while an incorrect answer with relevant work may
receive partial credit.
1. (20 points)
(a) Find the constants a,b,cso that the vector field defined by F= (x+ 2y+az)ˆı+
(bx −3y−z)ˆ+ (4x+cy + 2z)ˆ
kis conservative.
(b) Is F=xyˆı−zˆ+x2ˆ
ka conservative vector field? If so, find a scalar function f
such that F=∇f.
(c) Evaluate RCF·drwhere Fis the vector field in (b) and Cis the curve x=t2,
y= 2t,z=t3from t= 0 to t= 1. You may leave your answer as a sum of
fractions.
2. (25 points) Geoff, the proprietor of Land O’ Lake Ice Cream Company, is testing a
new machine for pouring ice cream into the cartons for sale. Given a set of axes in
which the carton is the cuboid region {(x, y, z) : 0 ≤x≤4,0≤y≤2,0≤z≤2},
the machine is pouring the ice cream in such a way that the distribution of Crunchy
Frog BitsTM in Geoff’s revolutionary frog-flavored ice cream is given by the equation
δ(x, y, z) = 2+ z2[bits/in3]. Megan, one of Geoff’s intrepid taste-testers, has scooped
out a lump of ice cream from the top of the carton in the shape of the hemisphere
{(x, y, z) : (x−2)2+ (y−1)2+ (z−2)2≤1, z ≤2}.
(a) Sketch the ice-cream carton with the hemisphere removed. Label your axes!
(b) Calculate the total number of frog bits in the whole carton (before the scoop is
removed).
(c) Show that the volume element dV is unchanged under translation of coordinates
—i.e. u=x−x0,v=y−y0,w=z−z0, where x0, y0, z0are constants.
(d) Use the result in (c), and another coordinate change, to set up the integral
necessary to calculate the total number of frog bits in Megan’s scoop using the
most natural set of coordinates. (You do NOT need to evaluate the integral.)
3. (20 points) Calculate the integral of f(x, y, z) = 2zover the wedge in the first octant
cut from the solid cylinder y2+z2≤1 by the plane y=x.
4. (15 points) Convert Z1
−1Z√2−y2
1
x
x2+y2dxdy into polar coordinates (but do not eval-
uate it).
