Math 263 Practice Test 3 - Lafayette College

Math 263 Practice Test 3

Name: You must show ALL of your work in order to receive credit. If your scratch paper shows work that leads to your solution,

please turn in in inside the test. Otherwise, dispose of it yourself.

1

1. Label each statement True or False. You may earn a bonus point by correcting a false statement. (a) The line integral of the function f (x, y) over the curve C represents the area of the surface below f (x, y) above C. (b) A vector field is a function that assigns a vector to each point in its domain. (c) Every vector field F has a potential function f so that f = F .

2. Given the vector field F = yi - xj, graph the vectors at the points (2, 1) and (-3, -2).

2

 3 2 1

3. Evaluate the integral

xyz2 dxdydz. (12 points)

0 -1 0

3

 8 2 1

4. Evaluate the integral

0

3 x y4 + 1 dydx. (12 points)

5. Rewrite the integral

z dzdydx in spherical coordinates, where D is the region in the

D

first octant bounded between spheres centered at the origin whose radii are 1 and 2. (12

points)

4

6.

Write

an

integral

in

polar

coordinates

for

the

volume

below

f (x, y)

=

x y

over

the

triangular

region in the xy plane with corners (0, 0), (2, 2), and (0, 2). (12 points)

1 1 1-y

7. Write an equivalent integral to

f (x, y, z) dzdydx so that x is integrated first,

0 x0

followed by y, then by z (you do not have to evaluate the integral). The region given by the

bounds is graphed below. (12 points)

5

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