APPM 2350 Final Exam Summer 2007
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, talking horses, calculators, or electronic devices of any kind are NOT permitted.
Note that this exam will be weighted as 150 points. Please start each problem on a new
page. Good luck!
1. (22 points) Consider the following integral:
2
Z
0
√4−x2
Z
−√4−x2
√x2+y2
Z
0
dz dy dx.
(a) Sketch the region that is being integrated.
(b) Convert the original integral into cylindrical coordinates using the order dr dz dθ
but do not solve.
(c) Convert the original integral into spherical coordinates using the order dρ dφ dθ
but do not solve.
(d) Solve one of the integrals.
2. (22 points) Consider the velocity field F(x, y, z) = x2i+y2j+z2k. Assume this field
is flowing through a surface which is a cube bounded by the coordinate planes and the
planes x=a,y=a,z=a.
(a) Find the flux through the surface by direct calculation.
(b) Verify the flux in (a) using a theorem from calc 3. Clearly state the theorem used.
(c) Find circulation around the right surface panel by direct calculation.
(d) Verify the circulation in (c) using a theorem from calc 3. Clearly state the theorem
used.
3. (18 points) Consider the region Rin the xy-plane bounded by y=x+ 1, y=x−1,
y=−x+ 5, and y=−x+ 3.
(a) Sketch the region.
(b) Use the transformations x=1
2(u+v) and y=1
2(u−v) to sketch the new region
Sin the uv-plane.
(c) If the density of the region Ris given by f(x, y) = e(x+y), find the mass of the
region by using the transformations given in (b).
