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THIRD HOUR EXAM, MATH 223
Instructor: Kris Green, April 10, 1997

1.

Multiple Choice (5 points each):

(a)

The area of the region shown can be computed correctly using which of the following integrals?

$$\int_0^2 \int_y^{(y+2)/2} dx dy \int_0^2 \int_0^2 dx dy$$

$$\int_0^2 \int_{2x-2}^x dy dx \int_0^1 \int_y^{(y+2)/2} dx dy$$

(b)

Which of the following vector fields could represent the gradient field of the function f(x,y) = x²/2 + y?

(c)

Which of the following vector fields best matches the verbal description below? ``The vectors of $\vec{F}(x,y,z)$ point toward the origin and have a magnitude which decreases as the point (x,y,z) moves farther from the origin.''

$$\vec{F}(\vec{r}) = -\vec{r}/r \vec{F}(\vec{r}) = \vec{r}/r$$

$$\vec{F}(\vec{r}) = -\vec{r}/r^2 \vec{F}(\vec{r}) = \vec{r}/r^2$$

(d)

The line parameterized by x(t) = 3-2t, y(t) = 1 + t, z(t) = -5 - 3t is parallel to which vector?

$$3\hat{i} + \hat{j} - 5\hat{k} -2\hat{i} + \hat{j} - 3\hat{k}$$

$$-3\hat{i} - \hat{j} + 5\hat{k} 2\hat{i} - \hat{j} + 3\hat{k}$$

2.

To find the centroid (physical center) of a two dimensional region R we evaluate the integrals

$$x_c = \frac{\int_R x dA}{\mbox{area of }R} \qquad \qquad y_c = \frac{\int_R y dA}{\mbox{area of }R}$$

Find the coordinates (x_c, y_c) of the region shown here.

3.

For the integral shown below, first sketch the region of integrations and then switch the integral to cylindrical or spherical coordinates, whichever is appropriate. Do not evaluate the integral.

$$\int_{-3}^5 \int_0^{\sqrt{25-x^2}} \int_0^5 xyz dz dy dx$$

4.

A particle is traveling in a force field along the path

$$\vec{r} (t) = t^3 \hat{i} + 3\hat{j} + (t^3/3 - t)\hat{k}$$

for t > 0.

(a)

Is the particle ever traveling purely horizontally (ie. parallel to the xy plane?) If so, when?

(b)

If the particle has a mass of 2 kg and the position is measured in meters, use the equations $\vec{F} = m\vec{a}$ to calculate the force acting on the particle as a function of time. (Note: $\vec{a}$ is the acceleration of the particle.)

(c)

If the force field is turned off at t = 3 seconds, find the parametric equations for the resulting motion of the particle.

(d)

How far does the particle travel in the first 10 seconds after the field is turned off?

5.

The path of the earth around the sun can be parameterized (roughly) by

$$\vec{r}_e (t) = 3\cos(2\pi t/365) \hat{i} + (4 + 5\sin(2\pi t/365)\hat{j}$$

with the sun located at the origin. The comet Hale-Bopp is also orbiting the sun along the path (completely made up)

$$\vec{r}_c (t) = (10 + 12 \cos(2\pi t/1000) \hat{i} + 3\sin (2\pi t/1000) \hat{j}$$

where t is measured in days for both motions.

(a)

On the same axes, with the sun located at the origin, draw the paths of both the earth and the comet. Label the point t= 0 for each, and indicate the direction of the motion.

(b)

Using vector addition (or subtraction) find the parametrization of the comet's path asseen from the earth. (Be careful...draw the vectors on your diagram above to see how to do this.)

(c)

Using the vector found in the previous part, what algebraic condition must be satisfied for the comet to hit the earth? Do not try to solve the equations, but explain your reasoning in one or two sentences.