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THIRD HOUR MAKEUP 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 area of the shaded region below.

(b)

Find the average distance to the origin of points in a disk of radius a that is centered at the origin.

2.

For the integral

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

(a)

Draw the regions of integration.

(b)

Rewrite the integrals as a single integral by changing into polar coordinates.

(c)

Compute the integral.

3.

The Cosmic Egg can be modeled as the region between the lower hemisphere of a sphere of radius 5 centered at the origin, and the surface z = 25 - x² - y² (all distances are in parsecs.) The density of the Egg is given by

$$\delta (x,y,z) = 5 - \frac{x^2 - y^2}{x^2 + y^2}.$$

If the egg has a mass of $3000\pi$ then it will be capable of hatching a being that will grow large enough to eat the Milky Way galaxy. Are we in danger of this happening?

4.

Calculate the flux of $\vec{F}(x,y,z) = (2x + 1)\hat{i} + (y - 3)\hat{j} + (4-z)\hat{k}$ out of the closed cube with corners at the points (2,0,0), (-1,0,0), (2,0,3), (0,1,0), and (0,-2,0).

5.

(a)

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

(b)

The Blob is attacking Springfield nuclear power plant. It has no definite shape at any point in time. It swallows a small source of radioactive fuel that emits radiation along the vector field $\vec{F} = \vec{r}/ r^3$. Calculate the total flux of this vector field out of the Blob.