1. A field biologist determines that the population density of dwarf
carnivorous moose in a particular region is roughly given by
dwarf carnivorous moose per square mile. Approximate the number of dwarf
carnivorous moose in the region
using the partition given by the lines x = 10 and y = 20, with midpoints
of each rectangle as representatives (xij*, yij*).
2. If the Earth is taken to be a sphere 3963 miles in radius and the
atmosphere is 600 miles thick, write a triple integral in spherical coordinates
that would produce the total amount of ozone above the northern hemisphere
given a function
for the density of ozone per cubic mile.
3. Show that the Jacobian for the transformation to spherical coordinates
(, , )
is .
4. Evaluate the integral .
5. Set up limits of integration for
if the region E is bounded by x = 0, y2 + z2 = 9,
and x = 10 - z. You need not work out the integral.
6. Evaluate the integral
7. Show that the center of mass of a beautiful purple disk of any radius,
centered at the origin and with constant density, is at the origin.
8.Show that the area of the part of the plane z = ax + by + c that
projects onto a region D in the xy-plane with area A(D) is .
9. Buffy is a calculus student at Oklahoma State. She says "Like, our
professor told us that when you use that Fubini thingy, like, all the little
limits switch around, y'know? But, like, y'know, I figured out that it's
the same when you do
as when you do ,
so, I mean, I guess it's okay to just switch, like, the dz and the dy,
or whatever, y'know?"
Is Buffy right, or are there corrections or limitations that should
be made to her statement?
10. A truncated paraboloid is formed between the surface z = x2
+ y2 and the plane z = a for some positive constant a. To what
depth should the paraboloid be filled with water in order that the water
have exactly half the volume of the whole solid? (Yes, you need to work
out the formula for the volume of the paraboloid even if you remember it
from the problem set.)
Extra Credit (5 points possible):
Show why the Jacobian of the transformation x = u(x,y), y = v(x,y) is .