Chapter 14.3 Practice Problems - math.drexel.edu

Chapter 14.3 Practice Problems

EXPECTED SKILLS:

? Be able to convert rectangular double integrals to polar double integrals, including converting the limits of integration, the function to be integrated, and the differential dA to r dr d.

PRACTICE PROBLEMS:

1. Consider the region R shown below which is enclosed by x2 + y2 = 1, x2 + y2 = 4, y = x and the x axis.

Fill in the missing limits of integration: f (x, y) dA =

R

2

f (x, y) dA =

f (r, )r dr d

/4 1 R

f (r, )r dr d.

For problems 2-6, evaluate the iterated integral by converting to polar coordinates.

4

16-x2

2.

00

x2 + y2 dy dx

32

3

3/ 2

9-x2

3.

x2 + y2 2 dy dx

0

x

243

8

1

2

2x-x2

4.

xy dy dx

00

2

3

5. Evaluate (x - y) dA where R = {(x, y) : 4 x2 + y2 16 and y x}

R

112 2

3

6. Evaluate

e-(x2+y2)

dA

where

R

=

{(x,

y)

:

x2

+

y2

3

and

0

y

3x}

R

1

1-

6

e3

7. Use a double integral in polar coordinates to calculate the area of the region which is inside of the cardioid r = 2 + 2 cos and outside of the circle r = 3. 93 - 2

8. Use a double integral in polar coordinates to calculate the area of the region which is common to both circles r = 3 sin and r = 3 cos . 5 3 3 - 84

9. Consider the top which is bounded above by z = 4 - x2 - y2 and bounded below by z = x2 + y2, as shown below.

2

Use a double integral in polar coordinates to calculate the volume of the top.

16 8 2

-

3

3

10. Consider the surfaces x2 + y2 + z2 = 16 and x2 + y2 = 4, shown below.

Calculate the volume of the solid which is inside of x2 + y2 + z2 = 16 but outside of x2 + y2 = 4.

32 3

11. Calculate the volume of the solid which is bounded above by z = 9 - x2 - y2, bounded below by z = 0, and contained within x2 - 3x + y2 = 0.

405 32

3

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