Prof. Porter’s
MAT251 Multivariable Calculus Sample-
Third Exam

1. Really take
some time and tell me HOW multivariable calculus can be used in your field of
study.

2. You are
asked to design a building that is over 15 stories tall with a square base and
a sloping flat top with one corner the highest and one corner the lowest.
Assume that each story is at least 10 feet. Identify the coordinates of at
least three roof corner points, give the equation of the function for the roof,
and then give the double integral that could be used to calculate the volume of
the building.

3. Show the
triple integral that could be used to find the volume of the building from the
last full story to the top of the highest corner.

4. Because of
gravity, the density of the building is higher at the base than the top. The
density function is given by the formula δ(x,y,z)=100/(z+10)
pounds per cubic foot. Give the integral that can be used to find total mass of
your building.

5. Give the
integral that can be used to evaluate how high is the center of gravity? A
building is considered unstable if its center of gravity is higher than the length of the
shortest side. Is your building unstable?

6. To cut
corners (literally), you are asked to make the design of the building perfectly
round at the base. Using the same roof design, what would the integral that
yielded the new volume of the building look like?

7. You wonder
if the volume of the cylindrical building would be easier to find in different
coordinates. Give the integral for the volume if it was changed to cylindrical
coordinates. Do not solve, but is it easier?

8. Whenever you
change from one coordinate system to the other for calculating the volume,
there may be changes in
the dV part. Set up the determinant that can be used to show that
dv becomes _{}when changing from rectangular to spherical coordinates.

9. You decide
to accent you building with a pure gold cover to the roof. Give the integral
that would tell how much surface area has to be covered with gold.

10.
Extend the ideas of triple integration to reduce this
quintuple integral into a triple.

_{}

11.
Give and graph
a conservative vector field.

Find
divF and curlF

Give
_{}

Or

Find _{} if **F**(x,y,z) =
<4x,2,2y> and **G**(x,y,z) = 2x**i** – y**j** +z**k**

12.
Find the
integral under f(x,y) = x+2y and over the curve y=x(x+1) when x is between 1
and 2

Or

If
C is defined by x = 2cost and y = 4sint for 0<t<π/4

Find
_{}

13.
How much work
is done when you move in a strait line from (0,0) to
(1,2) through the conservative vector field given by the potential function is
xy?

Or

Find
the exact value of _{}using any method if

**F**(x,y) =
<2xy,x^{2}+cosy> and C: **r**(t)
= <t,tcos(t/3)> for 0<t<π

14.
How much work is done in problem #13 if you follow
the path that forms a unit square centered around the
origin? Justify your answer.

Or

If
C is the unit circle centered at the origning and **F**(x,y) = <x,y> then

Find
the exact value of _{}

15.
Give a non-trivial example of Greens theorem and
solve it.

Or

Find
_{}if C are two cirles of radius 1 and 2 centered at the origin.