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THIRD HOUR EXAM
MATH 223
Instructor: Kris Green
April 3, 1998
9:00-9:50

1.

For each of the following, set up, but do not evaluate, a double integral in an appropriate coordinate system.

(a)

Find the average distance of points in the shaded region to the y-axis.

(b)

Find the area of the region bounded by the graphs y = x² and y = 2x - x³.

2.

For the integral

$$\int_{-2}^0 \int_{1 + x/2}^{3 + 3x/2} xy dy dx + \int_0^4 \int_{1 + x/2}^3 xy dy dx$$

(a)

Draw the regions of integration.

(b)

Rewrite the two iterated integrals as a single integral by reversing the order of integration.

(c)

Compute the integral.

3.

Mr. Spock detects a cloud of interstellar gas that occupies a region W of space. The density of the cloud is

$$\delta (x,y,z) = 100 - z^2.$$

The region W is the region bounded by the surface x² + y² - z² = -1, between z = -10 and z = 10. (x, y, z in astronomical units (AU), and density is in grams/cubic AU.)

(a)

Write down, but do not evaluate, an integral to compute the total mass of the cloud, using Cartesian coordinates.

(b)

Choose a better coordinate system and rewrite the integral in this coordinate system.

(c)

Mr. Spock claims that the cloud has a mass of 1,642$\pi$ kilograms. Is he right?

(d)

THIS IS A VERY VERY BAD TEST QUESTION.

4.

Calculate the flux of $\vec{F}(x,y,z) = \vec{r}$ out of the closed surface formed by the intersection of z = 25 - x² - y² and z = 9 in two ways:

(a)

Set up, and evaluate the flux integral directly.

(b)

Use the divergence theorem to compute the flux.

5.

(a)

State the divergence theorem. Be sure to use complete sentences, include all necessary conditions, and define all symbols used.

(b)

For the vector field $\vec{F}(\vec{r}) = \vec{r}/r^3$ (note: div$\vec{F} = 0, \vec{r} \neq \vec{0}$) a student states the following:

``For the surface S being a closed cube of side length 2 centered at the origin with W the enclosed volume, we have $\int_S \vec{F} \cdot \hat{n}dS = \int_W \mbox{div}\vec{F} dV = \int_W (0) dV = 0$.''

What is wrong with this statement (in complete sentences)?