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Instructions
- This stage of the exam is due in class, Monday, November 23, 1999.
I will not take late exams in my mailbox unless you have specifically
discussed it with me.
- Do this stage of the exam on your own paper. Do not put more
than one problem on the same page. Please remember to
staple your work before you come to class.
- Organize your work in an unambiguous order. Show all necessary steps.
- You may use a calculator on this stage of the exam where appropriate, but be sure to show the
work leading up to your calculations. Do not use your calculator for
vector operations.
- You may use your notes and your textbook on this stage of the exam.
- Do not discuss any stage of the exam with any other person besides
your instructor before all stages are complete and handed in. If you have
any questions, please ask your instructor.
- Good luck!
- 1.
- (15 points)
- (a)
- Sketch the region of integration in the xy-plane for the
integral

- (b)
- Evaluate the integral above.
- 2.
- (20 points)
- (a)
- Sketch the region of integration in the xy-plane for the
integral

- (b)
- Reverse the order of integration and evaluate the integral above.
- 3.
- (20 points)
Consider the solid lying above the plane z=0 and inside both the
sphere x2+y2+z2=16 and the cylinder x2+(y-2)2=4.
- (a)
- Sketch the cross-section of this solid in the xy-plane.
- (b)
- Set up a double iterated integral in polar
coordinates to find the volume of this solid. (Do not evaluate.)
- 4.
- (20 points)
In 1994, the Discovery Channel sponsored a scientific expedition to
study Loch Ness. One of the things that the scientists studied was
the amount of biological matter (plankton, fish, etc.) in the loch.
Their objective was to decide whether there was enough food available
in the loch to support a population of large predators.
- (a)
- Based on satellite photographs and sonar readings, the
scientists are able to approximately describe Loch Ness as the region
bounded by the equations z=0, y=x2, x=y2, and x+y+z=0,
with the restriction z<0.
Sketch top and side views of Loch Ness based on this description.
- (b)
- Suppose that the scientists were able to come up with a function
that gave the plankton density at location (x,y) on the
surface of the loch and vertical location z in the loch. Using the
description above, set up an iterated integral that would give the total
amount of plankton in the loch.
- 5.
- (20 points)
Consider the solid inside the surface 3z2=x2+y2 and between the planes z=0 and z=4,
with density
.
- (a)
- Sketch top and side views of the solid.
- (b)
- Set up iterated integrals for the mass and center of mass of the
solid, using either cylindrical or spherical coordinates. For this
problem, do not use symmetry. (Do not evaluate. You may use the
arctangent function in your limits.)
- 6.
- THIS IS AN EXTRA CREDIT PROBLEM WORTH AN
ADDITIONAL 5 POINTS.
Consider the part of
the ellipsoid

lying above the plane z=1.
- (a)
- Sketch the region in the xy-plane over which this surface lies.
- (b)
- Set up an iterated integral for the surface area of this surface. You
may use any formula for surface area.
If the formula you use has a cross
product in it, do out the cross product. (Do not evaluate the
integral.)
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The translation was initiated by Joshua Holden on 11/21/1999
Up: Math 103 Home Page
Joshua Holden
11/21/1999